Though of course much of the shine has indubitably disappeared since Brexit.
I was there 2008 through 2013 - and in 2010 we rented a relatively new 1 bedroom flat, on the 2nd floor (of 13), overlooking the Thames, around the Docklands (east London, near the secondary financial sector) on a clean, safe estate with a concierge. This was £280 / week.
A decade later, and doubtless having spent quite some time to find a highly unpleasant flat in a ridiculously inflated part of town (Camden) and TFA are citing £375 / week for a shared bedroom & toilet.
Like any metropolis, London is immense, with huge variations in quality and cost -- everything is a tradeoff against how close you want to be to the center versus how long you want to spend commuting.
FWIW when we moved from London UK to Sydney Australia in 2013 we went from river views, 40 minute commute to city center in London at £280 (~ AUD475) / week , to 60 minute commute to city center, highway views, for AUD500 / week, for the same quality apartment.
Never move to Sydney if you can avoid it.
EDIT: Oh, one of the problems with high-rises around London is the fact the whole place is a few metres above sea level, and the ground water levels are insanely high. I had engineer friends who worked on the underground (tube) extension years earlier, who told me that if the bilge pumps stop working, tracts of the underground lines would be flooded within fifteen minutes. The Thames Barrier (worth a google if you don't know of it) is unlikely to be effective within 10 to 20 years. Plus they've got lots of real estate locked up by investors who simply never occupy it -- all of this does brutally wonky things to real estate pricing there.