What if we come up with an easy to manufacture action that people can use to make their own? There's not much point in using a full sized rifle action. Bolt handle acting as a single rear locking lug? According to my Bolt Thrust and Hoop Stress spreadsheet :-) the 7.62x25 at 36,259 psi would have a worst case bolt thrust of about 4000 lbs during a case head separation. A single 0.40x0.40" lug on a 0.50" bolt body made of 92 ksi steel would yield at 17000 lbs bolt thrust. Normal safety margin on commercial actions is around 4x so that works out. With a 0.125 action thickness where the lug is bearing on it would give 0.050 sq-in contact area compared to an AR15 at 0.069" so it should be ok for a lower pressure cartridge like this. Hoop stress on a barrel with 5/8-20 tenon thread would be 107 ksi - similar to a 62,000 psi 0.473" case head cartridge in a full sized bolt action. At this size it would need to be made of centerfire rifle materials: a proper 4140 or 416 steel barrel and action. 4140 at 32 HRC hardness is 135-ish ksi yield. This action could be as small as a rimfire: 3/4" to 1" OD on the action. Should be able to make it fit on a Savage MKii footprint!
(For those who are not familiar with yield strengths of different steels, here are a few comparisons: Mild steel 36,000 psi or 36 ksi, Grade 5 bolt 92 ksi, Grade 8 bolt 130 ksi)
I think a Howa 1500 Mini in 223 would be your best bet if you wanted to start with a commercial action. A magwell adapter for TT33 magazines should be doable in the space available. I don't have metric change gears for my lathe, though. The Howa threads are metric - M26-1.5 I think. And the Howa Mini isn't exactly a cheap starting point but it is a good gun! Then if you get tired of 7.62x25 just put all the 223 parts back on.