As above if it's accredited by the BPS then it will cover all the necessary core content required for graduate basis for chartered membership (GBC).
LSE is primarily a specialist institution in the social sciences so will undoubtedly focus more on that angle of the subject rather than on e.g. the scientific or clinical sides of things. So I imagine they will focus more on things like psychological approaches rational choice theory and decision sciences, etc, beyond that core content (however the BPS requirements make up a pretty significant part of most psychology degrees so it's really more in terms of the advanced optional modules that you'll see the differences, so mainly in third and second years).
Note that other than LSE, many psychology courses do require or prefer at least one "core" science (i.e. maths/physics/chemistry/biology); some may require two (but often will accept psychology as one of the two, and sometimes other things like geography - but usually still require at least one core science). So I would recommend taking at least one of those core sciences, since as well as giving you more options, I understand it will prepare you for the course better than many alternatives. I've seen people say in particular A-level Biology and/or A-level Maths are the most specifically useful.
Depending on what you want to do afterwards, where you study the degree may not matter that much though. For example, if you plan to pursue a career in clinical psychology, the NHS doesn't really discriminate between individual universities and will just accept the accreditation meaning they are equal. However, as it's a very competitive field to get into, courses that include more clinical content and/or potential for clinical placements or attachments may be particularly valuable (and I'd note often these are not offered by "Russell Group" or similar universities, as they tend to focus more on the experimental science/research side of it - noting also the RG is a political lobbying group with no bearing on undergraduate teaching quality anyway).
I'd also note though that as indicated above, the name of the degree is basically meaningless - the important thing is the content. The Cambridge psychology course is also called psychological and behavioural sciences, and is renowned as being one of the more science heavy psychology courses.