Advice: Investigate and plan. First remove a section of the caulking to see if in fact the lavatory is indeed locked in by the tile. If not... faucet replacement is possible. Check availability and price of suitable replacement, or get estimate from plumber that is aware of your findings.
Locked in... It may also be possible to remove 1 row of tiles above the sink so it can be removed, repaired, remounted, tiles glued back into their original positions, grout/caulk to finish and ignore anything else you were thinking about replacing/repairing. You risk of breaking tiles that you may not be able to find a match for.
Decide what if anything you want to salvage. It would be possible to remove tile along the wall to the same height as the void left after removing (in pieces) the lavatory, and filling this space with other complimentary tile to form a back-splash the entire length of the new top. This would also allow some fudge factor in positioning the new lavatory for better appearance.
Remove all the tile. Be aware that damage to the underlying wall is likely and if it is lath&plaster may be extensive enough to warrant removing all the plaster and applying sheetrock over the lath.... yes.... over the lath unless the house was built late enough to have smooth planed dimensioned lumber.
Was the tub surround applied over the tile? This is another challenge but not an impossible task to remove tile and/or plaster up to and if desired over the top of the surround if the tile goes that far.
What shape is the tub and surround in?
Well as you can see, a simple washer is turning into an expensive labor intensive remodel.
Check the sink. That is easy to recaulk. ;-)
BTY... the faucet MAY have the old fashioned non-replaceable BEVELED seat (not sure when they quit making them) that you can ream a new face to with a special, relatively inexpensive tool if you can still find them. This could explain why the plumber cold not remove the seat, but don't go pointing fingers yet. You seldom ever see these old beveled seat fixtures any more.
Oh... Where is the toilet? It is normally against a wall so there is a place to mount a TP dispenser... like in the corner to the right of your sink. Exception would be where the original build had a base cabinet to hang a dispenser on, and then it is not uncommon to locate toilet between tub and sink.