Rent a drum sander and run it in line with the boards. You can take off a large amount of wood in short order depending on the grit of paper you use. Then use a sanding disc with a backer on a grinder for detail work where the drum will not go. We used 36 grit to bring some weathered wood back; probably took of over 1/4" off the top in that case! Total out of pocket should be around $200-$250 for everything.
Self answering to tell you about the results.
I started on the backpanel (which is normally out of sight so ideal for testing) with 3 different paint-stripper products to see which one would work best with this paint.
Used paint-stripper specifically for car-paint, regular household paint-stripper (both from the DIY store, made by Alabastine). And I tried a 20% solution of dichloromethane which I had leftover from an earlier project.
All 3 worked pretty well, but the regular Alabastine stuff got the paint of better, without the need to really scrape at the residue. (Given the fine detail in some places on the furniture I really wanted to avoid scraping/heavy sanding if at all possible.)
Some remnant stuck, especially where the underlying wax had been very thin or worn away. But from the test with dichloromethane I noticed that that stuff had worked exceptionally well in similar spots.
So I tried dichloromethane as a 2nd application to those spots where the Alabastine hadn't entirely worked. That turned out to work great and got nearly everything off.
So I ended up with making to passes: First the Alabastine and then the dichloromethane.
I then gently warmed the wood with a heat-gun and used a rag soaked in methylated spirits to draw out the remaining wax. Had to do that 2 or 3 times to really bring the wood back to its raw state. This also opened up the grain of the wood which I considered a good preparation for the new layer of wax I wanted to apply later.
The end result was quite good. Here and there a small amount of paint was left stuck in a deep scratch, but I could carefully scrape those out with the point of a knife.
A sanding with fine wire-wool smoothed the wood, without destroying any fine detail on the woodwork.
The wood ended up a bit uneven in coloring, but I will be able to smooth that out with colored wax when I re-wax the furniture later this week.
All in all it was far less of a nightmare than I originally expected. I am guessing the wax underneath the paint was my savior. If the paint had gone straight on the wood, it would have been much more difficult to get off.
Best Answer
If it was paint from the walls, I'll take a stab in the dark and guess that they used the cheapest possible latex paint to get the place ready to sell. Try scraping at it with a credit card, starting from the edge of the paint.
If it starts coming off, use the credit card to pick up a handful of cheap plastic putty knives and go at it. Most latex paints don't stick particularly well to enamel unless it's been pretty severely abraded, and cheap plastic putty knives won't damage it (metal will, or at very least leave marks).