Without Challenger, the Shuttle does probably see a lot more flights. 1985 had seen 8 Shuttle flights, 1986 was to have seen 2 in January alone (Challenger being the second). There were if I recall right a number of DoD missions and other flights scheduled for Shuttle that were switched to other launchers because of the return-to-flight delays and safety concerns, some of those stick around and launches like them keep coming.
However, this rate was coming with compromises to safety, with schedule pressures being allowed to outweigh good working of problems. It's possible some kind of other event occurs--perhaps a re-entry tile failure like Columbia. If it doesn't, there's going to need to be more Shuttles. Probably the spare-parts Orbiter (Endeavor OTL, may get different name here) comes to be, and Enterprise may be fitted for flight. OTL it was studied, but Endeavor was cheaper. If the turnaround times and flight rates require more shuttles, then Alt-Endeavor and Enterprise may both see the flight line alongside Challenger, Discovery, Columbia, and Atlantis.
My guess is a continuation of the ramp up, construction and fitting of at least one additional Orbiter to relieve fleet pressures (probably Alt-Endeavor if only one). Eventually, sooner rather than later, a major disaster. However, in the mean time, several missions fly without the RtF delays: the Galileo Jupiter mission, the Ulysses solar study mission (both scheduled for 1986, IIRC), Hubble (on track for October 1986, still with bad optics undetected). Magellan may fly to Venus earlier--I can't find data on when it began construction, so it's 1989 launch may be the earliest it was available.