What are my chances, UCL?

You should try reading their entry requirements...
https://www.ucl.ac.uk/medicalschool/undergraduate/mbbs-admissions/entry-requirements/entry2015.pdf

"Our standard conditional offer for 2015 entry: grades A*AA at A-level to include Chemistry and Biology (with the A* grade being in either Chemistry or Biology), plus a pass grade in an additional fourth subject to at least AS-level."

UCL are also pretty keen on the BMAT. A good BMAT score used to guarantee you an interview, I'm not sure if it does now but I'm sure you can find it written somewhere.

Have read many times, but not unlike others, it mentions little about how weighted each aspect of one's application is.

Good to know about BMAT though, cheers!

Well the only things they weight are what they put out there. There are no secrets!! They like people with a non-traditional extra subject, and they actually say that. I don't know if Psychology is really that much of a contrasting subject, but it's not one of the 4 they list as the traditional ones. The reason I quoted it is because it seems you're already outside the standard offer - they're asking for A*AA and you've got AAA.

It's a competitive course so really you want A*AA. If you miss one of their stated requirements or standards, then there's no secret weighting to that, it's out there on paper.

But maybe you've got better predicted overall A2 results. They generally use the easiest possible route to selection, which means that reading your personal statement would take them hours whereas excluding you on the basis of insufficient A Levels, insufficient BMAT score or if everybody has good enough A Levels and BMAT then discriminating on the basis of a contrasting subject. Then probably read the Personal Statements after they've excluded as many applications as they can without taking up hours and hours of time. That's just my guess but I think it's quite logical.