The short answer to your question is that British units have been reorganized and reformed to a similar degree to the US Army. Both the UK and US have seen periods of expansion and contraction in their militaries. For the UK, today their army is considerably smaller than it once was therefore, some active British regiments are amalgamations. Take for example The Royal Scots Dragoon Guards. This unit is an amalgamation of The Royal Scots Greys and the Carabiners. Instead of disbanding regiments with lengthy histories the UK Ministry of Defense chooses to combine regiments in order to maintain their history. This has the effect of maintain the lineage and history of units that would otherwise be not existent. The United States also maintains units that trace their lineage back to the very beginning of the country. The 3rd Infantry Regiment of the US Army traces its lineage back to 1784.

The slightly more complicated answer to your question is: that while both Armies have been reorganized and units have been disbanded or reformed throughout their history the US Army underwent a significant structural change that makes unit histories slightly harder to follow than the British Army. In the late 1950s the United States Army transition from a regimental centric system to a brigade centric one. The United States Army disbanded regiments as a tactical unit (with a few exceptions) but maintained regimental histories and lineage. In 1984 this was expanded to include combat support and combat service support units to form what is known as the United States Army Regimental System. Effectively unit histories are maintained through regimental affiliations despite the fact regiments do not really exist. For example, the 2nd BDE of the 4th Infantry Division contains the 1st and 2nd Battalion from the 12th Infantry Regiment (1/12 IN and 2/12 IN to use Army shorthand), the 1st Battalion of the 41st Infantry Regiment (1/41 IN), and the 3rd Battalion from the 61st Cavalry Regiment (3/61 CAV). These units each maintain unit histories and linages through their regiments rather than the Brigade. This means that 1/12 IN and 1/41 IN may share common history during the last 10 years, but 1/12 IN traces its lineage back to 1861 while 1/41 IN was organized in 1917. US Army Brigades and Divisions do maintain a history as well however most of these begin just prior to the 1st World War.

In summary, both the United States and the British Army have attempted to maintain unit histories and lineage despite increases and decreases in Army size and changes to structure. The British have done so in manner that consolidates unit histories and makes it easier to follow regimental lineages. The United States Army created a regimental system that keeps the majority unit histories and lineages in a defunct unit type that is separate from the actual command and control structure. This makes it harder for a lay person to track, but still maintains unit lineages throughout the history of the US.

Sources:

US Army Regulation 870-21: https://history.army.mil/html/forcestruc/docs/r870_21.pdf

US Army Unit Histories: https://history.army.mil/html/forcestruc/lh.html

History of The Royal Scots Dragoon Guards: https://www.army.mod.uk/who-we-are/corps-regiments-and-units/royal-armoured-corps/royal-scots-dragoon-guards/

Well, there is truth in both cases of regiments staying alive after war time as well some regiments folding after a war. The reductions to the military's numbers occur during peacetime when a war has ended and no threat is looming about. The Household divisions which include the Coldstream Guards among others falls differently in that manner. They are to be the monarch's own protection and regiments. I would not say that they did not have reductions as I have some pension records of former Guardsmen that left out of choice due to reduction, and they were given a pension for it. The dissolving of regiments has two reasons in most cases: Disbandment from the colonel or peacetime. Many regiments only rose to service for war and would be disbanded after the war.

One thing I want to point out is that Divisions are different than regiments. Regiments consist of one or more battalion. A battalion is 500 or 1000 men depending on the time period or strength of the regiment. Several regiments make up a brigade, and several brigades make up a division. Divisions are logistical in nature. They exist to keep things organized within units. The idea existed in the 18th century, but not quite what it would turn out to be later in the 19th and twentieth century. A twentieth century military historian might want to pop in on this, but divisions were mostly wartime. An example I do know is the divisions of the British army during the battle of Balaclava in 1854. These divisions would only last the campaign. The divisions that became established during the World Wars era would have entirely different logistics behind them for numbering and organization.

The British disbanded and reformed regiments in the eighteenth century. During the Seven Years War the numbers of regiments went up to 124. The British army did not have 124 active of full regiments but that is where the numbering system stopping in the Seven Year War era. I should explain the British regiment numbering system before I go on further. Until 1751, several of the regiments either just had names or they had a Colonel's name attached to them. For example, before the 10th had its number it was the Earl of Bath's regiment, the 24th former Sir Edward Dering's, and so on. Later in 1782, many of the regiments would get county or region names to them. The 24th also became the 2nd Warwickshire regiment, the 6th regiment was the 1st Warwickshire - but neither are called the Warwickshire regiments usually. It is just were they are from. In 1751, the Royal Warrant called for:

No Colonel to put his Arms, Crest, Device, or Livery, on any part of the Appointments of the Regiment under his Command.
No part of the Clothing, or Ornaments of the Regiments to be altered after the following Regulations are put in execution but by Us, or Our Captain General's permission

The colors of the regiment of course had to have the Great Union flag in it but also

In the Centre of each Colour is to be painted, or embroidered, in Gold Roman Characters, the Number of the Rank of the Regiment, within the Wreath of Roses and Thistles, on the same Stalk; except those Regiments which are allowed to wear any Royal Devices, or ancient Badges, on whose Colours the Rank of the Regiment is to be painted towards the upper Corner.

They have become numbered and have down away with the Colonel's name attached to it. At this point, the Royal Warrant recognizes the "Old Corps" of the Army which has roots in either the late 17th century or earlier in the 18th. Many of these have Royal in the names, establishing them as Royal regiments which allows them to wear the royal blue facings on their coats. Regiments such as the 7th, 8th, 21st, 18th, and 23rd. They had proven themselves in battle to earn the royal blue facings. One tidbit, the 1751 mentions the Highland regiment which will be numbered the 42nd that year becoming the 42nd Black Watch. They get their Royal Highlanders name in 1758. I will add that numbers had already been a thing for the British army before 1751 but became the official way to designate a regiment. Numbers came by seniority or by the dates they were raised. So the lower the number the older they are. Usually!

Remember how I said some of the regiments numbered all the way up to 124 before in the Seven Years War? Well, some of the upper numbered regiments did not last for more than a year or two since they may have formed very late war and some formed as Invalid regiments. Invalid regiments took sick, wounded, and infirmed soldiers out of active regiments. They did not do it for all soldiers but for the ones that were recommended into them, they still got army pay and would be relegated to garrison duty back in Britain. Mostly garrisoned in forts to recuperate. Not many went back to active service. It would give them a chance to gain a pension out of service since you either needed to serve twenty years or get really hurt in service for a pension. Anyways, back to the disbanded regiments. After 1763, the army reduced many of the established corps and disbanded regiments raised for the wartime service. After the Seven Years War, King George and the War Office looked at the budget for the army and likely had a fit. Later in the 1760s he decreed that there will no longer be more than 70 regiments on the roll. Many of them not active and stationed all over the Empire from Ireland, America, West Indies, and even Africa. So the regiment numbers stopped at 70.

That is until war broke out in 1775 in America. The very first new regiment established in the British army was the 71st Regiment of Foot or Fraser's Highlanders in 1775. So the number they got was already used back in the Seven Years War (From here on out it will be labelled 7Y War) and was completely different. Simon Fraser who raised the regiment had a regiment back in the 7Y War - the 78th Fraser Highlanders. However, Fraser could not just skip the numbers between 71-77 for nostalgic reasons and 71 would at least give them seniority when other new regiments would be raised. When the war picked up and France and Spain got into the war, the British army grew and raised more regiments. Man numbers reused and recycled into new regiments completely different from the old ones. Coincidentally, the 77th and 78th Highland regiments of the 7Y War era also became Highland regiments during the American War for Independence.

Once the war ended in 1782, again many regiments disbanded. Some took until 1784 when they to wait for prisoners to return or the full regiment to be grouped before they could begin the process. The Siege of Gibraltar had also ended which meant that the regiments used in Europe would soon be disbanded as well. The 71st Highlanders would be among the many to be disbanded.

Peacetime led to disbanding of regiments in both the British and American army due to the wariness of having standing armies. For British it was fear of corruption and a repeat of Cromwell. The population wanted professional armies but did not want tens of thousands standing around in garrison. This is what led to reductions and disbandments. It was to save money as well. Paying for armies costs a fortune. During this time, Britain also had militias and fencibles for home defense. A more part time army and only relegated to stay within Britain. Rarely, did they leave the borders. The American army was also transitioning to a peacetime force after 1783 as well. The Continental armies disbanded that year to form the new American forces. The 3rd Old Guard is still active and has roots from the First American regiment in 1784.

To go back to the 71st, this number would be reused a few times more until 1881 when the Childers Reform came to be. I will explain that later on. The 71st post American War of Independence, was yet again another Highland regiment but raised by a different colonel. Initially called the 73rd Highland regiment they would be designated in 1786 as the 71st. 71st MacLeod's Highlanders would remain so until they became the 71st Highland Light Infantry in the 19th century with the Childers Reform they changed their name again to be the Highland Light Infantry.

The British army kept the highly distinguished and seasoned regiments overall. Some of them survived the reforms in 1881 in some fashion and kept their identity through most of the twentieth century. However the amalgamation in the later twentieth and early twenty first century changed the regiments once again. Some of the amalgamations happened after 2000 which is not yet within the 20 year mark to discuss. The current regiments or battalions may have its roots from those in the past but some are not quite the same. While not dissolved, they had been reduced and renamed into different battalions. The 7th Royal Fusiliers which had been established in 1685 became the Royal Fusiliers (City of London) in 1881 and by the middle of the twentieth century, they amalgamated with other fusilier regiments to become the royal regiment of fusiliers. Regiments have dissolved and reformed many times over the years just as the American armies have done the same.