The first solution will be to build an ISS. Perhaps, as part of some commercialization of supplies to the moon, supply flights with the participation of N-1 could be made. The biggest change would be the return to using RP-1 in LR-87 engines.
My idea is that ITTL the OTL Shuttle-MIR would instead be Freedom-Moon Town
Zarya having a nuclear meltdown with a death and a radioactive shuttle would stop any replacement from being flown, think of what Chornobyl did to most people's idea of nuclear power. Nuclear power in space as a whole is dead, nobody would dare fly a reactor outside of small ones for lunar missions (RTG's)
Basically, IF the Soviets launch one the US of A would likely have a major issue with it, same with most other countries, the late 80s was a period of easing tensions and newfound cooperation, a nuclear station would cause this to be shattered, the US would likely condemn it and maybe even put sanctions on soviet exports or imports (Pepsi shipments)
I don't mean to sound like a crazy person but public sentiment is based on examples, very few will actually learn the "how it works/happened aspect". Nuclear reactors OTL have largely stopped being built due to safety concerns and nobody wanting them built in their area
New York City for example bans nuclear-powered ships from entering the harbor or bay around it, the HMS QE was the first to enter in a long ass time due to this reason
people picketed Cassini's launch due to a few kg of plutonium, Apollo 13 had to do a course correction to put the RTG in the Mariana Trench
Basically, my idea for how TTL plays out is that the American Freedom Program works in conjunction with the N1 and Soviet/Russian program, TTL's ISS analogue is the International Moon Base or IMB, a joint base with Soviet/Russian and US modules, the N1 is kept around to keep Russian engineers employed and launches 1 manned mission every year. The Americans launch one a year as well and this creates a permanent manned station with 6-month crew rotations, Cargo is flown with the landers and later commercial companies, likely TTL's Delta-4 analogue launches the landers