Your shirts sound fine, and most places would be more than happy with them. I've seen that 'no shoes, no shirt, no service' thing in various countries but it isn't a major thing in the UK because our climate doesn't really encourage lots of shirtless, shoeless action; however I have seen it in a few seaside holiday resorts.
The one thing that I would mention is that some places have a 'no jeans' policy. That's generally the more upmarket restaurants, nightclubs and gastropubs; but your average pub or student bar will have no problems with you wearing your usual t-shirts, jeans and trainers.
So keep the shirts for evening trips to classier establishments, and if you do wear jeans, it's safer to wear black ones that just look like plain black trousers. Most places with a 'no jeans' policy would also forbid trainers; you'd have to wear neater shoes. Also, many establishments will refuse entry if you wear any kind of sportswear.
Above all else, just be observant. Look at what your friends are wearing; walk past the pub/club and look through the window or at the queue outside to see what people are wearing; or best of all, Google the venue first to see what their dress code is and dress accordingly. UK dress code can be interpreted as follows:
- 'Casual' = blue jeans & t-shirt; trainers or sandals are fine
- 'Smart Casual' = your long-sleeved shirts (no tie, open collar is fine) and chinos or black jeans (no blue jeans); neat but casual shoes (e.g. loafers, boat shoes)
- 'Eveningwear/Business/Lounge' = suit & tie; smart, lace-up, patent leather shoes
- 'Black Tie/Semi-Formal/Formal' = tuxedo; smart, lace-up, patent leather shoes
So Smart Casual would be a safe bet for most places you'd want to visit as a student. Oh, and even if an establishment doesn't state its dress code on its website, almost all clubs have photo galleries where they like to show off how much fun their patrons have had. These photos will show you what people usually wear in there...
