Pros: London is wonderful to walk around and being central gives you so many options for entertainment, dining, travel (hop on the Eurostar etc)
Cons; lack of inside space, outside space and parking space, noise from the revellers in the evening and the bin men/commuters/workmen in the morning.
Flats are inevitably leasehold/freehold which comes with it's own set of issues. If the cladding/water tank/ roof needs sorting this can come with hefty price tags and months of meetings to sort. AirBnB flats below/above/beside you are often there to party and are loud & disrespectful. Renters often leave their furniture in the hallways/lift/stairwell if they need to vacate and don't want to take it with them.
Getting a plumber/electrician/carpet fitter out to central London can be a nightmare (& pricey) as they can't park/have to pay congestion charges etc.
Getting furniture in and out of London flats again is a nightmare - large pieces of furniture simply won't fit in the loft/stairwell and you can't necessarily park anywhere close so end up carrying it past people drinking/dining on the pavement.
Homeless people/drunks/drugged up people often sleep /piss / puke in your doorway, plus if people don't check the communal door is shut behind them they also get into the building and sleep/shag/deficate in the halls.
Deliveries go missing (unless you have a concierge) as they've been "delivered" to the communal entrance hall.
People buzz you at 2am as they've hit the wrong doorbell/fancy pressing the buttons for a laugh.
Communal smoke alarms go off in the middle of the night as someone has decided to smoke in their flat/burnt their pizza/not cleaned their oven in years and you can't turn them off.
I mean it's great, but there are definitely downsides!