I buy most of my casual stuff (e.g. lunch with Mom/the girls, biz casual at office, getting on a plane level) from Seasalt because the quality is just outstanding and the pieces are finished correctly, last forever, and usually have some really nice/surprising detailing that makes everything look pulled together.
I will note that one of their tanktops in cotton has wrinkled pretty heavily and just doesn't hang quite right on me, but everything else has always worked great. Definitely wait for a sale though - they tend to have them every 12 weeks or so and its possible to pick up some really good deals.
Otherwise yeah, agree with you, finding decent quality basics is challenging here - I am still wearing some cotton tops I brought over in 2014 that I got for like $8 each at a regional department store, but cannot find anything close to similar for the same quality short of paying 3x that price. Uniqlo does in a pinch, but its not quite the same weight. Ditto Marks. I refuse to go to Primark because I can't handle the sensory overload.
Ah - you could try SimplyBe - one of their sister brands. I got this cotton tank dress last summer for CHEAP - maybe £12? from some weird German label through them and that thing is comfy, of the weight I was looking for, the quality I was used to, and it wears great. I putter around the house/neighborhood in it instead of shorts. Unfortunately I keep losing it in the closet because its black and looks like a tanktop. SimplyBe is for the larger ladies, but I know they have smaller sizes, just run under a different 'shop'. And I think you want to look under the Basics range. I see they have a 3-pack of cotton tanktops for £13 right now - maybe I will get some and try them out as some of the Uniqlo ones are on their last legs. Just be aware that I think they have a 14 day return policy.