Does the London tube have good facilities for carrying bikes?

No. Non folding bikes are not allowed on most tube services in central london. The lines that do allow it don't allow it during rush hour.

So how do all the bikes get into central London?

Not sure I understand your question. I think people cycle them in…

You can take them on overground trains, though that's kind of restricted at rush hour. I imagine people mostly either cycle from their homes or leave the bike near a station.

Lots of stations have large cycle racks. They just go from office to station and back again.

While there are certainly some areas like the City of London which are almost exclusively commercial/office, most of London, even the central parts, is a mashup of different land uses including residential. Not everyone commutes in from the suburbs.

Folding bikes are sometimes not allowed

No, but folding bikes are pretty common in London. Those can be taken on trains at any time whereas normal hikes are technically only allowed on off-peak trains.

Nope, they are not allowed on the underground network unless they are folding bikes.

Outside peak hours they are allowed on the sub-surface lines (Circle etc) as those have larger trains and simpler emergency evacuation.

It's still not something you'd want to do very often. I think I did it once with a new bicycle that I'd bought from the other side of London, where the route home would have been long and completely new to me.

Map of allowed lines: https://content.tfl.gov.uk/cycles-on-public-transport.pdf

For non-UK people, particularly American readers, the City of London, and inner London in general has zero public parking. It feels like even saying 'zero' is being pretty generous. Every street is restricted in some way (no parking, or exorbitant parking meters), and there are decreasingly few car parks/parking lots.