They probably don't care but:
The fatf puts a limit of £1000 on cryptocurrencies which came into force in the UK.
The UK also has a limit of doing £15k/transactuon before "potentially" requiring aml info. The government have kept everything pretty legally grey in this space so it's hard to work out what they actually care about. I haven't gone over about £12k from coinbase to bank. Realistically the onus is on the exchange as its accounts are the ones taking the funds and the ones who should be blocking illegal activities.
From my own experience, yeah they haven't started caring (not sure if Brexit has had an impact) but definitely not to the people I know am trading with. I've mentioned before that now exchanges are getting a bit titchy in the EU, requests for Proof of Funds now coming to me from 3 exchanges. Varying levels of difficulty in satisfying their requirements, but have to note that these come from crypto deposits rather than cash outs (which they freeze until PoF is satisfied).
Like you said, the onus is on the exchange, and for the banks, if the exchanges have satisfied AMLD5 -- and this usually means at most the bank calling up the exchange (as the originator of the tx) rather than you as the recipient.