My husband and I were called for 'flu jabs a fortnight ago and had them at our excellent surgery.
We were both invited (by text) to book anti-Covid boosters at a local leisure centre. My husband booked his for the same day as his 'flu jab (but had it in his other arm).
I have had problems with 'flu inoculations in the past (a topical reaction - I grow a bump the size of an egg!), so was advised to wait eight days from flu to booster. I am therefore having my anti-Covid booster on Friday.
As I did not immediately book my anti-Covid booster (following a text from NHS England), it was assumed that I was dilly-dallying and they texted me twice further. Only when I booked a slot did the texts stop. I thought this was actually great and that their system was on the ball.
Incidentally, GP surgeries do not know when they will receive an allocation of 'flu jabs. They are sometimes only given a couple of days' notice of their arrival and have no control as to what /when jabs will arrive, be they 'flu or anti-Covid vaccines, so please do not blame your GP!