All 3 are great places for engineering id probably say Bath has the slightly strongest reputation followed by Warwick for engineering, UCL is a great university but not particularly well known for mechanical engineering (that said i don’t think you would struggle with postgraduate study or graduate career opportunities either).
I broadly suspect a lot of UCL graduates end up working in London and this is then reflected in graduate salaries (and employers forced to pay more due to living costs), I also suspect far more UCL graduates choose to pursue careers in financial services or similar whereas Warwick & Bath likely see much higher rates of students entering professional engineering employers.
You would be absolutely fine at all 3, it’s important to note whether you do postgraduate studies or a professional job what the recruiters will care about is you as a person not the university you went to (yes university can reflect competency however all 3 here are close enough that the assumed competency will be equal).