I’m very pleased you raised this because I am in exactly the same position (and I daresay plenty of others are too).
Recent BBC article says:
“The government said it was working to ensure that the system would work for people across the UK, while also considering how to verify the vaccine status of non-UK residents.”
So it seems they are working on it. Let’s hope they have successfully done so by 9 October.
In the meantime, I have taken the advice of the other responder to your post and asked Nicola:
Will let you know what she says….