My opinion is that the Montanas would probably end up going into reserve shortly after the end of the war, and are disposed of during the early to mid-60s.
Although the Montanas would be better as battleships for fighting other battleships, because of somewhat greater firepower & considerably better armor, assuming that the evolution of naval warfare & geopolitics remain roughly the same post WW2, with the carrier replacing the battleship as the primary capital ship, and the battleship being demoted to a supporting vessel, useful as a heavy escort for the carrier primarily valued because it mounts a lot of AA, and the occasional shore bombardment, then it'd make more sense to retain the Iowas in post-war service.
Compared to the Iowas, the Montanas are slower, thirstier, and require notably more manpower to run (which also makes them more vulnerable to post-war budget cuts) while out of all the battleships it had, the USN considered only the Iowas to be truly compatible with fast carrier task force operations. Furthermore, unless that third set of locks for the Panama Canal get built in that ATL as well, the Iowas would have better strategic mobility, since they don't have to go the long way around to move between the Atlantic & the Pacific.
Also of note is that in the battleship concepts that the USN's designers and planners mulled after the cancellation of the Montanas IOTL (just concepts that didn't even reach the preliminary design study stage), the idea for that hypothetical next battleship design was an improved Iowa, which would be enlarged to allow for better seakeeping and torpedo protection (both considered shortcomings due to the design compromises necessary to get the needed speed and maintain the standards of protection from the prior class while being constrained by the 45,000 ton displacement limit allowed under the escalator clause and being restricted to a 108.25' beam due to the need to fit through the Panama Canal), as well as improved deck armor, a stronger AA battery, & better sustained sea speed under full-load & long time-out-of-dock conditions.
As far as modernization potential goes, they would have had about the same potential as the Iowas, but, again, their higher operating costs would be a strike against them- indeed, during the 1980s modernization, the operating costs of the Iowas were such that the USN seriously considered not reactivating the main battery to save some money, until the absurdity of that was pointed out. (Incidentally, from statements on Navweaps by Dick Landgraff & Bill Jurens, the Iowas were actually more ship than optimal for that modernization, & the Alaskas, had they still existed, would have been 'just right' from the size/operating cost perspective.)
Had any been built, I don't think the Montanas would have been likely to have seen action beyond the end of World War II- perhaps if budgets are more generous TTL, Korea, but really, to have the Montanas stick around after WW2 when they're not as suited to the post-war battleship missions where the carriers are now the center of the fleet, you'd probably need to come up with an ATL where post WW2, there's a hostile nation with a powerful enough surface fleet, including a battleship force, that's enough of a threat that a battleline has to be kept around to deal with it. Of course, such a world would probably be radically different from OTL.