Using the Thames as your "corridor" for selecting a place is definitely a good strategy. Numerous modern developments along both banks, ranging from smart to high end luxury. Twickenham in the west to Greenwich in the east is about 20 miles. Some of the locations feature attractive marina developments. The river itself offers great views. Remember that a north bank development won't be so great for views over the famous London skyline, which is mostly on the north side of the river, but this is compensated for by greater sun through the day.
There is a fast and frequent river bus service, the Thames Clippers, which serves the central-Greenwich river section very well, but which is Mon-Fri only on the westwards section as far as Putney only.
Choice could be really hard. Some areas of London can be pretty quiet after 5pm and at weekends. You might have other factors concerning what you will be doing in London that might be good drivers for selecting a borough.
might be a bit quiet for a month or so, however if you hold off until after restrictions, you may want to visit:
London (stay in Notting Hill - this would form the bulk of your time)
Oxford, Windsor - (day trip)
Lake District (stay in Keswick for a couple of days and get your walking boots on)
Pop over to
Scotland (Edinburgh and the highlands)
if you can get across to Ireland hire a car and drive around, starting from Dublin (stay there for perhaps a day, 2 max), down towards Wexford, Waterford and Cork drive up to Galway and back to Dublin. stopping off for a stout or two
you would probably have seen all you need to see, and will give you a good idea what the UK is about
just make sure you travel as fast as you can from london to the lake district. there are some really grotty places in between!
and from the lake district to Scotland, just close your eyes
oh and officially southern Ireland isnt in the UK, however i'm assuming you want to come back from your travels, so ive neglected Northern Ireland completely.