Back again with these sorts of questions? Still not sure where you are heading here. You say you are not doing an educational thesis, and in another you said you were worried about customs or immigration control but they would not care either. You might also do better if you went and made a big thread rather than having people repeat themselves every time; again NES history, much less UK NES history, is not really a particular focus of the site and it is mostly older members that were around that will be filling you in here.
Short answer
Anybody that cared to buy one. Most toy manufacturers do wholesale selling of things, and there were still quite a few independent shops throughout all that. The toy company known as Mattel (a few of these were later or after but Barbie, Polly Pocket, Match Box, Hot Wheels, He-Man, Fisher-Price are under their umbrella) were the ones handling distribution in Europe for it and they would deal with just about anybody.
Nintendo were also not the only game in town -- the "crash" was mostly a US (possibly also Canada but I know less there) affair. The UK and Europe had all sorts of things going on throughout all this so while the NES was a big player it was far from the only one (Sega was a somewhat bigger player in Europe than it was in the US, the commodore 64 and later the Amiga was no small fry, the amstrad, bbc micro and all the others were fading by this point but still had a presence), not to mention UK retail at the time worked rather differently to US retail today.
Longer answer
Supermarkets had not quite yet got in on the act; that might have even been as late as PS2 in some cases, and even then only some popular games. Even today I normally only expect to find games, online credits and maybe a controller or two.
Toys R Us would have had some in the UK, indeed such places were probably the main place a lot of people went.
EB Games (what eventually became Game, and bought up Gamestation) arrived in the UK in 1995. Not sure what the Rhino Group (their path in) were doing during this as I am not sure if any were around me.
Woolworths might be a place to look here. I don't recall what they did for the NES offhand but they were around for the games on tape and floppy discs eras.
Virgin and HMV, both retailers more known for music, had a few games in. Virgin's parent company had a game division (Virgin Games during most of your time frame, becoming Virgin Interactive towards the end) that were active and often well respected in game dev/publishing, though as far as I can tell never did anything for the NES despite being around at the time (
https://www.giantbomb.com/virgin-interactive-entertainment-inc/3010-176/ https://www.mobygames.com/company/virgin-interactive-entertainment-inc https://segaretro.org/Virgin_Interactive ).
Argos will probably be worth looking at here. They do catalogues too which is probably good for you here if you can find a digitised one.
https://issuu.com/retromash/docs/argos-no36-1991-autumnwinter has a whole bunch going back many years, though possibly a bit spotty around the late 80s which a pity.
I am not sure what goes for high street chains (this was still before the real rise of out of town warehouse efforts). Dixons, Hughes, Currys, possibly Index, Toymaster might be worth a look but they are more a group of independents,
Blockbuster (the video rental place) in the UK eventually sold games and consoles, and they certainly rented them during this time frame, but I am not sure they sold consoles and games at that point save for a few ex rentals.
https://ebid.s3.amazonaws.com/upload_big/5/6/2/1380931576-31771-891.jpg
mentions a few as well, many of them long since defunct (and possibly even forgotten by the populace at large -- you might have to find some fairly old people who remember Allders) and a few of them looking like independents as well (Gloucester Toy Shop not really being a chain). Quite a few of those are also mail order or catalogue shops which is nice if you can find their catalogues.