hi i was pleased to see your post - as i am in a somewaht similar situation - i have dual indian & eu citizenship and also considering going to londong to spend some time there ( through i do realise probably not the best time) - though i am still awaiting my passpost which i have applied for.i also work in banking - though i am still in employment
having lived in uk as a student before i found NHS is very accomodating - probably you could get a travel insurance for a month / 2 to play safe for untill you get registered with the NHS.
you can easily open your bank account once you get there - just carry some pounds (cash) / traveller cheques for use in the interim
i noticed this website for shared accomodation classified; you could try it out & maybe even book before you get there. also lots of temp traveller accomodation in central london - [generator hostels is quite conveniently located but there are plenty more]
London Classifieds - Accommodation and Jobs in London, UK
Id be interested to keep connected to so we could beenfit from each others experiences - though i suspect i would only be making any moves in june / later. if this works for you we connect on email?