As I understand it, there are two vaccination streams (I have said this before) One is where you are invited by your GP practice. You may not go to their premises for the jab but that stream is organised by the GP's who staff the vac centre, employ the extra staff and so on. In this case you don't get both appointments at once.
If you go down the national stream, you book online, either after getting a letter or because you have been told that you are in a group who can book without getting a letter. The vaccination base and staff are employed and managed directly by the NHS and they are using pharmacies, mega hubs and hospitals. You book both vaccinations at once.
I live in an area with a very high percentage of frail elderly and my local GP group who have ganged up to provide the jab service have asked us healthy old farts to use the NHS stream to get vaccine so they can concentrate on the frailer folk. Wherever you get the jab "tells" the other lot so I can see my vac details on my GP record.