Second undergraduate or reapplying to prestigious universities, e.g Oxford, LSE, UCL

Hello!
I am currently in my second A Level year, sitting exams soon! I'm applying for Psychology BSc, however, recently I've become quite unsure of my choice, I feel that I am likely not going to enjoy taking the stereotypical route (post grad / masters + work experience) to becoming a qualified Psychologist. As such, I have been looking at options to apply my future degree to other areas, with my area of highest interest being in Law.
With this, would anyone be able to inform me whether prestigious universities (Oxford, Cambridge, LSE, UCL etc.) would be less likely to give you an offer (to a Law course specifically) if you are applying as a second undergrad, as well as a with a psychology first undergrad?
Would it be better, with higher chances of receiving an offer, if I were to simply reapply?
(for context I have a decent academic record for prestigious universities, not insane, as I had some pretty significant exceptional circumstances during my GCSEs, however I am predicted A* A* A A for A levels). As a little add on, would they consider how prestigious your first uni was in their decision (quite a silly question, I understand).
Apologies for the long-winded question!

Are you applying for a second full three-year undergraduate degree? If you are interested in Law primarily as a career, why not consider a Law conversion course? Similarly, if you are interested in Law as an academic subject, why not do Law as your first undergraduate degree?
It may be worth considering especially funding options. Funding works markedly differently for second undergraduate degrees so you may not be able to afford to do it without separate funding.

This is another one of my worries; I would be applying for a second full undergraduate degree (as I am interested in it both academically, alongside career prospects). I was thinking about the Law conversion course, however, I saw a few people say that it might not be valued as highly by top law firms, or your options may become limited, in comparison to taking either undergraduate or postgraduate degree? (I could be wrong about this!) Furthermore, I'm extremely lucky , as I would be doing my first degree as a home fee status student in Scotland (so fees are heavily reduced) as well as having parents who are fully supportive in financially supporting academic endeavors (like taking a second undergraduate / post graduate prospects). As such, my main concern is looking at the most prestigious / most valued academic record to have (if I were to go into Law).