Because the low countries offered ideal naval launching grounds for an attack against England. The low countries' ports are dangerously close to the center of British commerce located in London and the Thames estuaries, allowing them easy access to raid Britain's trade. Furthermore, while England's coast facing France are full of cliffs and other high places providing natural defensive advantage, the east of England facing the low countries, particularly the area around Essex are flat, thus far easier to attack. Various river systems and estuaries runs through the low countries into the north sea which meant that enemy transport ships can shelter in those places relatively safe from Royal Navy in preparation for a rapid attack on Britain itself. Possessions of the low countries particularly by France also negates the Royal Navy's strategy of guarding western approaches to counter French fleet sailing out of Brest. Finally, the shipyards of Antwerp and Flushing are also capable of building additional ships to attack Britain with, providing another incentive to keep it out of French hands.

Source: History of the Royal Navy, A: The Napoleonic Wars By Martin Robson

As a Dutch student interested in "our" history, I would like to try and answer this question.

Let me start off with saying that the most important aspects of this are naval supremacy and trade. Militarily the possession of the Low Countries would not have been too much of a problem for Great Britain.

Since the Dutch Republic had revolted against Habsburg rule in the late 16th century, they had quickly become one of (if not the) the richest nations in Europe, the world's financial and trade capital, and the world's foremost naval power during the 17th century. Especially the latter was of major concern to the English (who had not yet unified with Scotland and Ireland at this point). During the course of the 17th century multiple Anglo-Dutch Trade Wars were fought, in particular after the passing of the Navigation Act. Although the Dutch were victorious in all these 17th century conflicts, the power balance had started to shift in the 18th century. Dutch naval supremacy ended, and that position became vacant during most of the 18th century, with France, Great Britain/England, the Netherlands and to a rapidly decreasing extent the Spanish and Portuguese competing for this position. By the end of the 18th century, this was first and foremost between France and Britain.

Now, Amsterdam did remain Europe's foremost financial and trade centre throughout most of the 18th century. However, as was the case with their maritime power, Paris and London started catching up, and by the turn of the 19th century especially the London Stock Exchange succeeded Amsterdam as leading financial capital.

If you sum these development up, you see that the Dutch were not too much of a problem for the British anymore on their own. The Fourth Anglo-Dutch War in the 1780s was essentially an easy win for the British agaist the Republic (the latter being diplomatically supported by France). However, and this is the key to the answer on your question, a combination of Dutch and French (+ allies, most importantly Spain) fleets and ship production could be a major threat to British naval power. And, as was and is traditionally the case for Britain, naval power was absolutely quintessential in the British geopolitical defence strategy.

However, the Dutch fleet got defeated separately in 1797, and Napoleon's naval ambitions were definitely crushed after Horatio Nelson's famous victory over a combined French-Spanish fleet at the battle of Trafalgar in 1805. After these two maritime disasters for Napoleon, British naval supremacy was a definite given all the way until World War II.

Tradewise the story is more or less the same: with the addition of the Low Countries to the Continental System fears (or hopes, depending on which side you were on) were that British commerce would be hurt severely by being denied access to the by then moderately powerful but still very wealthy traders in the Low Countries. But also here the Low Countries were not as important as they had historically been, and the British were not hurt too much by this trade blockade.

In summary: Britain was indeed (very) weary of the acquiry of the Low Countries by the French given the former's historical naval and trade power. However, their power had been waning since the early 1700s, and the Dutch defeat at the Battle of Camperdown before Napoleon's final and definite naval defeat at Trafalgar meant that the majority of this threat had ultimately completely vanished.

-- I would recommend reading up on the Anglo-Dutch trade wars if you are interested in the topic of the Dutch geopolitical situation. Especially the Third Anglo-Dutch War is an interesting historical event if you ask me.

Martin Robson wrote a good book (A History of the Royal Navy: The Napoleonic Wars) in which the naval side of what I argue would be the answer can be found. I have to admit most of my knowledge on the Dutch 18th century comes from Dutch books unfortunately, and I have not yet found their translations online. I will give it a try later today.

Regardless, I find your side question about the Dutch geopolitical situation interesting and I actually cannot say for sure what the answer would be. I hope to see this further on in the comments.

I hope to have answered your question a bit!

The Low Countries remained a potent threat to Great Britain from the 1700s to the 1900s, due primarily to geography. British naval bases were typically positioned to meet the threat from the enemy of the day; for much of the 1700s and 1800s, this was France, while moving into the 20th Century, Germany became more prominent. Naval bases to meet the threat from France were placed along the British south coast, at Portsmouth and Plymouth. They could threaten any attempt by the French to gain control of the English Channel, and prevent French ships from their bases along the Bay of Biscay moving into the area. As Germany became a bigger threat, the primary bases for the Royal Navy move northwards, to Scotland. From here, the RN could control the North Sea, preventing the Germans making any threatening move towards the UK. However, this positioning left much of the British east coast uncovered by Royal Navy bases. This was exacerbated in the age of sail, when prevailing winds could pin British ships from Portsmouth and Plymouth in the Channel, while allowing an enemy to sail across to the east coast. In the 20th Century, the east coast was a vital shipping route. Ships that had crossed the Atlantic (or from the rest of the world) would moor at the major west coast ports, where their cargoes would be cross-loaded into smaller coasters for shipment to the east coast. This was a particularly vital lifeline for London. Any threat to this shipping was an existential threat to the UK.

The Low Countries have several excellent ports, at Rotterdam, Antwerp, Bruges and Ostend. Ships operating from these bases were excellently positioned to strike at the British east coast, and at the shipping lanes along it. As such, it was always a major priority of the Admiralty to ensure that these ports were always under the control of a power that was friendly (or neutral) to Britain. This included sending the Royal Naval Division to Antwerp when it became threatened in 1914. When much of Belgium fell under German control in 1914, the RN made multiple efforts to neutralise the German-held ports of Zeebrugge, Bruges and Ostend. Minelaying and bombardment operations were carried out throughout the period. The British Army carried out several offensives to attempt to capture the ports. There were several plans for amphibious attacks to capture the ports, with Operation Hush the closest to being carried out. In April and May 1918, the RN carried out several amphibious raids on Zeebrugge and Ostend, aiming to render them useless to the Germans. These raids did manage to sink blockships in the canals leading inland, but the ships were sunk in the wrong places to effectively block the ports, and they were soon usable. In the Second World War, the Low Countries still represented a naval threat, but the threat from German-held France was judged to be more serious; the Luftwaffe could threaten British air superiority over the Channel, but couldn't effectively reach the East Coast.

Am I really the only person here that doesn't know what a "Low Country is? No one else has asked this yet?