I don't know. My friend's DH got his MBA in England (they're American) and she and her children could legally stay there for the two years he was studying. I know there was a lot of rigamarole with the paperwork, but most of the MBA students had spouses and children with them. Her daughter even started public kindergarten there because they were all legal residents. This was three years ago. So, OP, you might want to check what the visa issues would be for the UK. You could probably go to the UK's American Embassy's website to check.
However, the PPs are right about your DH working. My friend couldn't work and it was *very* expensive to live there. They ended up going through their (very substantial) savings to fund her DH's tuition, plus the cost of the whole family living there for two years, especially because she wasn't bringing in any money. Let me emphasize again that the UK is very expensive and London is one of the most expensive cities in the world. If you've got *a lot* of money saved for tuition plus the cost of living, you could look into it.
I have a lot of experience with London as I did a post-doc there for two years and affordable housing is in short supply. You all would probably need to live pretty far out and pay a fair amount to commute in to wherever your university is (I'm assuming in the city center if it's one of the University of London colleges). My friend had a fantastic experience and are very glad they did it but it cost them their entire savings and, honestly, hasn't really paid off monetarily in terms of the jobs her DH could get with the degree.
Really investigate the visa issues and cost of living before you jump into this. A good place to start would be the university itself. They may have a foreign student office that could help you.