For now nothing actually changes, British citizens are still EU citizens, a London bank is still an EU bank.
In principle the country is "rapidly" negotiating a long term arrangement on everything for completing by the end of the year. This hypothetical "deal" would set up the relationship from 2021 onwards.
In practice there are two schools of thought. Maybe the intent is to get to the end of the year and accuse the EU of having somehow failed to achieve a deal (even though the EU was always clear that would take much more than a year) then break off with no relationship and suffer a huge economic down turn as a result. Maybe somehow that works politically? It's hard to see how, but it's not as though Brexit made sense anyway.
Or maybe the idea is to build up the tension and then announce we've belatedly discovered that it isn't possible to finish the deal immediately and so we'd like an extra say, two years to finish making a deal, kicking the can down the road. Who knows.
The UK actually left the EU on January 31st, 2020. We're still 'Europeans' (inhabitants of the continent of Europe) but no longer EU citizens.
You're correct, British citizens are no longer EU citizens (though there are court cases that dispute this) but all the freedoms they had as EU citizens continue until transition ends (currently supposedly at the end of 2020).
I just think it's amusing the idea that London has weathered the Fall of the Roman Empire, the Black Plague, the Blitz, and the Brexit vote, but what will finally dim its star is Twitter sanctioning WFH.