When you say, "go into neuroscience", what do you mean? Neuroscience is an academic discipline, not a job sector. If you want to go into academia, you would need to get a PhD (and possibly a masters along the way). You can do a PhD in neuroscience after a medical degree, but neither is exactly "being in work". There are some NHS BMS type roles in clinical neurophysiology (as an allied heath profession, rather than medical specialty) I believe, but these would probably expect an appropriate accredited undergraduate course and then structured training scheme.
Clinical psychology is another thing altogether, but my understanding is normally you need a BPS accredited undergraduate course, plus a masters and/or DClinPsych afterwards, or to go into a structured training scheme (maybe both?). Either way, it's unlikely to be much different in terms of experiences than medicine -> foundation -> specialty training except in the subject specific content. You're still going to need to do a lot of academic work and academic style training while working (and quite possibly some rather boring generalist work before you specialise more).
I think in both cases, you aren't really getting what you want from it; you're going to be going from 3-4 years remaining of your MBBS (2-3 of which will be clinical and possibly more engaging for you), to a 3-4 year BA/BSc + MSc/MA, or an MSci undergrad masters course, and/or a PhD or clinical doctorate (3-4 years again), plus some kind of structured training period (like you would have after graduating from medicine, although quite possibly more competitive to get into and with lower pay...). It's probably going to take a similar period of time to qualify in either of those non-medical areas as in a medical analogue of those (e.g. neurology, clinical neurophysiology, psychiatry - the latter probably being the shortest option).
might have some thoughts, since neuroscience is sort of an academic parallel to neurology in some respects (I think?). However I would note, I think ecolier has noted that they're changing the training for physicianly specialties to have a 3 year post foundation core training period, from 2, before higher specialty training. This might be even less appealing on the medicine front, although as above I don't think in terms of time frames it's much different than the alternatives realistically.