How do you install a replacement bathtub (plumbing is still good)?

Your question tells me that you need a professional plumber.

You should have access to the plumbing on the opposite side of the wall from the faucet and shower. Possibly a room adjoining the bath.
The shower head may not be accessible and that entails cutting through the shower wall. I hope not, but these are the possibilities.

Can the tub fit through the door, removal and the new?

If not, it has to be cut.

Also, it’s very likely that the configuration of tub base will not be the same as the existing one. If the configuration is smaller at any points along the front base, your flooring may need to be replaced, unless you have some spare to match. If the new tub is larger, adjustments are easily made.

In new construction the bath is often completed before the door frame is set.