Red Cat
Apologies if this is what you’ve done and it didn’t work.
Posting them through the cat flap
One person on one side and another person on the other side, both offering encouragement and praise and treats, and whenever possible holding them and physically posting them through to be received and praised. Only do one or two trips in either direction, and only if the cat isn‘t horribly stressed. Obvs. (I’ve also done posting them through with only one person and then coaxing them back through once they’re out. This sometimes results in a pissed off cat turning their back and humphing off for the rest of the day.)
Coaxing them through
Put them outside, prop open the cat flap with a stick or wooden spoon etc and coax them through with encouragement treats and praise. If they still won’t come through, leave them to it and see if they try.
Leaving them to it
Put them outside, prop open the cat flap and leave a trail of treats from just outside the flap, on the flap ridge somehow, and inside the cat flap. Give them enough time to lose patience. You might even have to leave the area for a while so they feel as if they’re out of options.
Show and tell
Sit down with them and demonstrate the opening and the closing, with soothing, to show that the noises are cat flap associated and normal in that context.
A combination of these has always worked for me…
(,..until now, because Not-stray point blank refuses to use the cat flap. She seems to have agreed not to shit or piss in the house in return for being let in and out on request. I have to try her againm and will report back. I suspect she had an unhappy start and it looks like the cat flap may have been one of the villains in her origin story.)
With two cats you’ll have to either work with one at a time, or hope one of them learns from the other, or something.
The last thing is that after you’ve done all the coddling and coaxing and teaching, leave them outside while you’re there and let them deal with it. They’ll either train you to always have to use the door, or you’ll train them to use their door key so they can come and go at will.
They’ve used the cat flap, they’ve used the hole in the wall, so they know the location and the process.
Is it something to do with the flap? Was it see-through before and not now? Is it much heavier?