In general, the consensus is 'don't do it': you leave yourself far too vulnerable.
It’s not something I would ever recommend doing - way too risky in my opinion.
It would be much quicker (and easier) to either try to qualify for ILR visa on your current visa route or to marry your UK partner, switch to FLR(M) and start the 5 years for ILR from there.
From a quick search online, if you were successful in becoming stateless and getting leave to remain, it looks like it would probably take you a further 7-8 years from now to qualify for ILR:
- at least 1 year to be approved to renounce US citizenship
- at least 1 year for the leave to remain application to be processed after renouncing
- another 5 years in the UK on two 30-month visas to qualify for ILR
- another 1 year after that to qualify for U.K. citizenship
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