Well, the USA still had conscription in place until the 70s, technically, it just went unused for much of the time. It would be easy to see a system under which, if enough volunteers were not forthcoming, places were filled up by selective draft even in peacetime. It's not quite the same thing as a conscript army, but with the US system and strategic position (emphasis on technology, no militarised borders, strong navy and air force component) it is very hard to see how a conscript army can be accommodated.
As for the Soviet forces, that would require a huge shift in doctrine right off the bat. It would actually very likely have done them good (and yes, of course voluntarism would be voluntary only to a small extent, with military service likely tied to all manner of goodies like higher education, vocational training, preferential housing and job placement, party candidateship and such). A smaller but better military (which would still be larger than NATO's) with a more reliable quality of recruit would cost less (the benwefits accrued to the veterans would only shift costs, not increase them) and would still effectively deter any invasion. I just doubt anyone in the Kremlin would see it that way.