All UK schools are regularly inspected and the results available for anyone to view. Schools are usually allocated by catchment - how close you are to the school, though some have specific boundaries to a catchment so someone less close, but inside the catchment boundary, takes precendence. It varies.
The postcode for LSHTM is WC1E 7HT. So if you go to OFSTED's website, you select "reports" and then punch that postcode in, with a 5 mile radius, saying you are looking for primary schools. I did that and got this.
Each report will tell you the social mix of the school as well as the achievements in the eyes of the inspectorate. OFSTED is very imperfect, but teachers I know say they're only out by one rank in either direction, as a rule, so opt for a Good or Outstanding, and rent somewhere in that catchment, and you should be okay. Though IMO nothing is better than a personal visit before you decide. A Satisfactory school I know of is a lovely place to be, and an Outstanding a results factory. Depends on what you want.
You need to be aware that very central London has a sharp split between state and private, so state primaries will have a lot of kids whose English skills are poor and will need a lot of support. While the cultural diversity will be fantastic, you need to make sure they have funding and facilities in place to support that diversity so all the kids learn well. Obviously supporting kids with various special needs takes up time and resources, as it should, so the schools need to be allocated those appropriately. A school with a specialist unit will often be a really good one.
Be wary of any OFSTED reports where the head has recently left. A head can make or break a school.