I don't mean to be rude, but you need to do far more research about the basics of owning and living on a boat before you worry about the finer points of which boat to buy. There are loads of boaters all wanting to be in and around London, and the place is basically full. They have grown in numbers over the last two or three decades, both encouraged by lifestyle articles in the media, and driven by the increasingly unaffordable cost of renting or buying in the capital. Official permanent moorings are few and expensive, so most London boaters are so-called 'continuous cruisers' engaged in an ongoing process of musical chairs shuffling about just enough to keep CRT's enforcement people off their backs. And many struggle to manage the movement, the need to fill water tanks, empty toilets, buy diesel and gas, generate enough electricity and not knacker the batteries, heat the boat etc., all while doing a 9-5 job in an office.
If you are in London 'several days a week' you won't have time to get any distance away in your non-working days.
Cambridge and Ely are a fortnight's travel away from London, via the narrow beam Northampton Arm. The Bedford-Milton Keynes Waterway is a pie-in-the-sky project which will take decades to complete if it ever happens. The only alternative route to the Anglian waters is via the sea or on the back of a lorry.
Sorry if that sounds brutal, but going starry-eyed and uninformed into boat ownership looks like an effective way of burning through your inheritance and ending up without a lot to show for it.