It took me some phone calls and some driving around, but I've found somebody nearby with a vacuum press and experience in veneering. I've read on the internet that you could do it yourself with all sorts of home remedies, but I have serious doubts about that. I'm not talking about some new varnish, but real new veneer.
The man told me to remove the old veneer, so I took the wood parts out of the oldest car, because the veneer had started to come off anyway.
The sides are in a different veneer than the front, and both veneers are attached to a backing.
Hard to see the veneer under this crackled varnish
I've used varnish remover here, more out of curiosity than anything else, because I'm going to use new wood anyway. I've only done this on one part. The wood chipped off easily without removing the varnish so why bother...
The wonderful walnut veneer on the glove box. On the right, some of the varnish has been stripped of with a special tool. You can see the varnish really brings out the patterns in the wood.
You can see the veneer around the doors is coming off by itself. Also note that the parts above the doors do have a veneer front in this car. In the 240 they do not, they're just varnished solid wood. Probably to save money?
No high-tech equipment is needed... just a scraper and - as always - lots of patience.