Mother got a letter from the NHS six days AFTER her appointment for the first jab
That letter could well be legit.
There are vaccination slots allocated to GP surgeries. The GP contacts you directly, usually by txt or email and occasionally snail-mail. Details are provided on how to book a slot at a limited number of supposedly more local venues.
And then there are the big vaccination centres such as the ExCel Centre and Epsom Racecourse. For these the NHS, as opposed to the GP, sends a letter inviting you to book a slot. However snail-mail only, no text or email. Trouble is that these letters are behind schedule and further delayed in the postal system. So it's entirely plausible that the NHS letter arrives after a person has already been vaccinated.
Judging by the BBC news last night it seems that the no-shows, at whatever the centre, account for up-to 20% of appointments. Probably some people book both and cherry pick the best time, but fail to cancel the unwanted appointments. That's the scandal because the no shows are delaying the roll-out and potentially wasting the vaccines.