The Finns and the Russians both used Mosin Nagant bolt-action rifles. Many Finnish Mosin were built using parts of captured and gutted Russian rifles. I myself own an M30 model from Tikkakoski of which the underside of the receiver bears the Russian Tula star.

As for most nations, remember a massive logistical difficulty: most allied nations had their own caliber. Americans had .30-06, the British had .303, the Russians had 7.62*54r, etc. The Germans used 8mm Mauser. So if they picked up Garands or SVT-40s or SMLEs, they would have a hard time keeping a proper stock of ammunition to feed them.

The Germans used a lot of captured armour. /u/clovis69 has already mentioned the Czech and French tanks.

They also made use of captured British tanks in North Africa and the Russian front (Valentines and Matildas), captured US Shermans and Russian T34s.

"Hitler's Great Panzer Heist" by Anthony Tucker-Jones is a great source of info on this.

Aircraft-wise, the Germans tried forming fighter squadrons on 3 separate occasions with captured aircraft: Czech Avia 534, French Hawk 75s and Italian Re2005s but gave up quickly.

Most captured aircraft were either handed over to training schools or used as transports but Germany did hand some French fighters over to Finland - Ms406, De520, Hawk 75 and Caudron 714s. The Finns also used captured Russian aircraft.

Source: "Foreign planes In The Service Of The Luftwaffe" - Jean Louis Roba

Ships are a bit more difficult to put into use but the Axis did manage it in a few cases.

[This warshipporn post is about a British destroyer used by the Japanese.] (https://www.reddit.com/r/WarshipPorn/comments/48axwk/sclass_destroyer_hms_thracian_1941_run_aground/)

After the Italian Armistice, the Germans seized some light craft - minesweepers and corvettes - and used them as escorts in the Mediterranean.

And something I only found out about the other day: The Croatian Puppet state formed a naval arm but the Italians objected strenously to any attempt to provide it with ships so the Croats ended up using ex-Russian small craft on the Black Sea (Armed trawlers mostly)

Source: Nigel Thomas - Foreign Volunteers of the Wehrmacht

Edit: Supposedly the Luftwaffe used captured Fortresses and Liberators to infiltrate USAAF bomber streams.

Obviously not small arms like in your question, but still related.

The German Luftwaffe had a special unit, Kampfgeschwader 200 (KG 200), that would refurbish and fly captured Allied aircraft. They did this both for testing for design secrets their engineers could use in German designs as well as to find performance flaws or tactical weaknesses that could be exploited by attacking German planes.

The Luftwaffe also employed some captured Allied aircraft, albeit on a very limited scale, as decoys to get close to unsuspecting Allied aircraft to attack them or create confusion and separate formations.

Did the US or UK ever use captured weapons? So far they're not mentioned here.

In addition to the captured armor simply given a new paint job and a Balkankeuze slapped on the side, there were also various conversions made, usually on the self-propelled gun side of things. For example, so many Soviet 76mm guns and ammunition were captured that a vehicle conversion program was started to result in the Marder tank hunter. The soviets did the same thing, such as the SU-85i based off a Panzer III.

Similarly a program was started in early 1942 when the M3 Medium first showed up. The 75mm ammunition was ineffective against the German panzers, due to some design failings. The solution was to use captured ammunition from the German 7.5mm gun (which was amongst an 18,000 ton supply dump capture), mate it to the American ammunition casing, and use that. It proved to be slightly more effective than even the improved US munitions, for as long as supplies lasted.

When Germany invaded and/or took over factories of other nations, they would re-purpose the captured factory equipment to produce German weapons (this is why you sometimes see Kar98k rifles produced by countries such as Czechoslovakia). However, countries such as Poland who were at war with Germany or countries like Belgium who produced weapons for commercial purposes continued to make weapons as the Germans came knocking on the factory doors. In those cases, the Germans would use the weapons already produced such as the Browning Hi-Power handgun (produced by FN of Belgium).

In terms of actual combat captures, the German soldiers did capture Allied weapons and would re-designate them. As mentioned before, the Germans in the Eastern Front sought after the PPSh-41 for its close-range effectiveness. Aside from firearms, the German soldiers would take Soviet combat boots and clothing for better warmth.