Swords started to become an archaism well after the Bow did. Primarily because despite all the merits it had, the Musket was never too efficient in a melee at fighting off multiple foes.

Swords stopped being common among infantry by around the Great Northern War with the invention of easily attached and utilized bayonets. These were cheaper and often able to be produced in a higher quality and quantity.

This never made swords stop being used wholesale however. Officers often had and frequently used their swords until around the period between the Crimean War and the Franco-Prussian War. (1850 - 1872). After this period they often times were still part of the Officers Dress uniform or regalia.

The usage of swords lasted quite long however in the tradition of Cavalry units until they too became an archaism on the approaches of the Second World War. Cavalrymen could never use the bayonet efficiently on horseback and therefore continued to refine and use the Sabers which became rather renowned for their shape and effectiveness in combat. Swords really fell out of usage near the end of the First World War.

Bows are a different story however as they became an archaism almost as soon as Muskets became easier to produce. The reasoning for this rather quick transition comes in a few factors. Ease of use was the biggest one, it was and is much easier to train an effective gunner than it is to train an effective bowmen. This allowed for nations to quickly replenish their numbers of trained gunners which could not easily be said of bowmen.

The bow had also remained a relatively unchanged concept for the last 1000 years prior to the invention of the early muskets. This is contrasted with the quickly transforming and evolving design of the musket. Even with these base factors the musket was a much safer bet and most states did the transition quickly.

The last time bows and swords was used against an exclusively musket based army was probably the Satsuma Rebellion in Japan which is the 1870's. This technically doesn't count of course because the rebels DID use muskets. However the commanders and men who were Samurai quickly resorted to the older alternatives once ammunition became scarce.

These men were defeated in two battles and most of it's participants killed. The bows and sword were of course deadly and efficient when used by the Samurai. But those Samurai could easily be killed by a barely trained Imperial Japanese soldier using a rifle.

After that Bows and Swords could of course be used but with each year they became increasingly scarce as the chances of success while using one dwindled significantly. By the time of WW1 swords were more of a decoration than a weapon and bows had been regulated to competitions.

As others have mentioned swords remained almost a universal sidearm up until 18th century, when they started to be phased out among the infantry so that they would not interfere with reloading. However they continued to be used by cavalry and officers.

As far as western Europe goes, bows generally stopped being used sometime in the 16th century. England was one of the last nations to fully phase out the bow, and during the 1544 invasion of Boulogne Henry VIII's army was still mostly traditional county levies armed with bows or billhooks while the French had mostly replaced their crossbows with arquebuses. Blaise de Montluc apparently thought it was sort of a novelty to be fighting English archers, even though much earlier in his carreer he had been in command of crossbowmen. He noted that early on his troops had quite a lot of respect for the bravery of the english, but after a number of skirmishes he concluded that this was a myth, stemming from the fact that the english archers had to get much closer than the french arquebusiers to use their weapons. He gives the example of one engagement between 150 french arquebusiers and pikemen against 200-300 english in which he ordered his troops to hold their fire until the english got within bow range, then deliver a single volley and charge, which easily saw the engilsh off. From the English side the perception doesn't seem to have been much better. According to captain Elis Gruffydd, "I never saw Welshmen or Englishmen so bad hearted or so unventuresome as I saw at this time. Not a single one of them would dare to go near where the handguns were shooting at us."

The second half of the century saw pikes replacing bills and guns steadily replacing bows among the english troops. During this period the attitude towards archery among many veterans continued to sour. Barnabe Rich in 1574 wrote that he still thought english archers could be effective as long as they didn't have to go head to head with enemy shot, but in his later works started to discount them completely. Reverend Harrison complained in 1587 that wherever the English archers showed up the French and Germans would bare their backsides mockingly shouting "shoot, English!" In 1590 Sir Roger Williams called English archers "The worst shot used these days" and Humfrey Barwick lumped the longbow in with javelins and axes as weapons only fit for "savage people or poor potentates." Then in 1595, Elizabeth's privy council finally declared the longbow obsolete and reclassified all remaining archers as unarmed men.

This doesn't mean that bows were completely gone at this time. Archers continued to show up to english musters well into the 17th century, and older weapons were still being used by some peoples such as the Irish. Barwick even claimed that he occasionally still saw archers or crossbowmen while serving as a mercenary in foreign armies. However these weapons were increasingly being seen by most as outdated or a symbol of backwardness.

Ironically in the 1570s, an English traveler living in Moscow named Jerome Horsey considered the Russian backwards because they were still using horse archers. In his opinion, 1200 scottish cavarly with pistols would be able to do more work than 12,000 Russian cavalry with bows.

"The commentaries of Messire Blaize de Montluc, mareschal of France" by Blaize de Montluc (1500-1577)

"A breefe discourse, concerning the force and effect of all manuall weapons of fire and the disability of the long bowe or archery" by Humfrey Barwick (1592)

"A briefe discourse of vvarre" by Sir Roger Williams (1590)

"A right exelent and pleasaunt dialogue, betwene Mercury and an English souldier" by Barnabe Rich (1574)

"Longbow and Hackbutt: Weapons Technology and Technology Transfer in Early Modern England" by Gervase Phillips

Gun Culture in Early Modern England by Lois G. Schwoerer

Elizabethan Military Science by Henry J. Webb