Our MH is a 4-berth (one double where the table/seats are and one double over the cab, which I would never sleep in because the headroom is very low, we use the space for storage). In terms of vehicle size, it's 5.2m long, and 2m wide, so it fits in a standard supermarket car park space. It has a lav, a shower, hot water (gas) heating (gas) and a fridge that works on gas, or from the leisure battery or van battery, or on electric hook up when on a site.
It's full height all along, so no faffing with lifting the roof. However, that excludes some car parks - 6'6" height barriers are surprisingly common.
Buying one is a minefield. Ideally, you'd want one with a recent "habitation certificate", so you know that everything works and there are no damp/leakage issues. I'd recommend joining a few motorhome forums and getting tips on what to look for. Diesel may also be an issue if you plan on using it in London, some sort of emission rule meant a lot of people in London had to sell older diesel vans a while ago. Also start looking at vans for sale online, and seeing how much they vary in price.
One of the big advantages of a motorhome over a camper is Britstops. You pay £30 for a book giving you a list of all Britstop members in the UK. They are places where motorhomers can stay overnight for free or for a nominal charge and we have stayed at some fantastic places for nothing. On a 10 day trip to Cornwall in March, we only spent 2 nights at a camp site. And we save a small fortune by being able to make a cuppa whenever we like, without having to pay rip-off Costa prices at service stations!
Ours was the best purchase we've ever made. We can go away for just a night whenever we feel like it, it takes less than 10 minutes to get ready because we keep most of what we need on the van.