I'll start off by agreeing with advice you have already received, The Indoor Market in Birmingham, Chinatown. There's one particular store that has a good selection of Thai vegetables. I think it's the one below. Obviously there's the other Chinese supermarkets very close by too.
The used to be a small shop in Cleveland Street Wolverhampton, the owner was from Singapore and stocked quite a few products from Thailand. I can't confirm this but someone said he now had a stall in Wolverhampton's Indoor Market which is by the outdoor market.
Pinoy Foods Stall 53/Indoor Market/50 Edgbaston St, Birmingham B5 4RQ