Tricky for this age, isn't it? When they're too small for men's but too big for children's. (Slippers are my problem atm)
My 13yo DS is not Inthe least fashion conscious, so I'm going off which shops worked for my DD at that age (so I apologise if the selection is off!)-
H&M for plain cotton basics, and the 14+ size fits smaller adults
River Island
Quicksilver is Roxy for girls, and the sizing is on the small side. It's listed in bands of age- 10, 12, 14 years, but like euro clothing this means 'up to 10', 'up to 12' etc. So my DD is 5'4" and slender, and has outgrown her Roxy age 14 tops. I always ordered online, and the clothing quality was reasonable (quiksilver is more expensive than H&M basics, but the things she had were nice cotton and lasted well, didn't get holes etc)
The online shop I used quite a bit was surfdome, and they will have the sale on right now. (usually children's footwear is heavily reduced on there, and I always get their backpacks from there too)
Another one for surfwear, vans etc is SkateHut.co.uk and again, their sale is usually v good for children (stock range is fairly small though, as it's an independent shop (physical shop).
Depending on his actual size, Zara is probably your best option for shirts- again, smaller sizing as it's European, but they have excellent quality cotton button-down Oxford shirts pretty much every season, usually in decent colours (my DS has navy, and light blue). They often have grandad-neckline and linen versions for summer/holidays.