OK, I believe you... but on mine, the surface is so hard that I can't imagine it being any kind of vinyl. On the wooden ones, it is indeed real veneer, and I believe the base board material needs to be that much thinner in those (I have a tatty pair of those as well, oak I think, waiting on me to decide on their destiny).
If you decided to go the re-vinylling route, you wouldn't of course be limited to wood imitation... there's everything, from lime stone to something Rich's wife would approve π
I myself try to preserve originality when ever possible, but as yours obviously can't be helped anymore, I certainly am not going to blame you for making anything you like out of them!
I think before during and after photos are in order when the time comes.
Yes, we love photos here! π