The calipers are handed, so I hope they've ordered the right one (or the left one... if you see what I mean). I expect it would cost more to have them rebuild it, and take longer, it's a few hours at least.
They will have to undo the hydraulic line to the caliper and let air into the system - but they should clamp the flexible hose next to the caliper to minimise fluid loss, fill the new caliper before fitting it and bleed it well when they have finished. I would still check that the pedal feel hasn't gone soft and complain if it has.
This is one of those areas that should really be part of the checks carried out at
service intervals, but it isn't. Same as the bonnet lock - give that a good squirt of teflon oil or even motorcycle lube every six months or so to prevent it seizing.
The complicated dual-action caliper is a common way to fill the Construction and Use regulation for a completely separate handbrake system. Some cars have a separate mechanical caliper with its own pads, and others a drum brake setup inside a large-diameter hub. They are all to some degree prone to seizing from the action of water/road dirt.
