What's it like driving in London city centre?

I've been driving for about a year now. I've only driven in and around Wolverhampton/Birmingham. The furthest distance I've driven was probably Wolverhampton to Milton Keynes. I was scared of driving on the motorway for a few months and went everywhere via the A roads but I'm confident on the motorway now.

Now I need to drive my nan to see her sick relative somewhere in central London, we'll probably stop in Essex as well to see another relative. Public transport is out of the question as she has a bad knee.

I'll probably take an experienced driver with me but she's never driven in London either. She's been to Cornwall, Wales etc

How different is driving through Birmingham city centre compared to London city centre? My car is ULEZ compliant, would I still need to pay for anything? Do you have to pay for going into certain zones?

I've passed my driving test, second time with the hardest test centre (Wolverhampton) in the UK (according to Google) if that makes anything easier?

The last time I went to London was over 5 years ago and that was with the train and we got around with Uber. It looked fine and I don't mind a bit of traffic.