They are as
@Best said, globe taps. They would more than likely have been put in by the Plumber using putty and paint, whilst the bath was stood on end and before it was installed. They will be difficult to remove but not impossible.
They are not available anymore due to regulations on back siphonage. The spout is within the bath and although you have an overflow which appears to be below the spout level, if it blocked, even partially, or has a flow rate less than that of the taps, then it is possible for the taps to become submerged in foul water. They are existing so don't need to be replaced retrospectively as far as I know.
Have you thought about what you will replace them with?
As best said, I would be tempted to repair them and spend some time on that. It should be possible for them to be re-seated, washer-ed and packed, as long as they are not corroded.