It's probably just as "locked" as any other time period, which is to say not at all. Recent SNW episodes have hammered home the idea that the present, past, and future are all somewhat malleable.

Voyager Before and After pretty specifically tells us that whenever someone jumps in time, the fact that they have done so, and the actions that they take, affect the timeline. Any time time travel is involved, we are seeing only one possible timeline, not necessarily the timeline.

Of course, it’s easier to use that reasoning in an episode like Before and After, because the time travel involved is backwards, but I think it’s reasonable to assume it’s true the other way, too. After all, the Kelvinverse isn’t identical to the Prime timeline even before the Narada arrived.

Did you edit the title in sync by any chance?

Just fixed that bug...

How likely is it that Discovery went to a mutable future, just one of many, especially with the Temporal Cold War, Carl, Q, Trelane, Janeway, the HMS Bounty, and any number of other temporally active agents out there in time? How locked in is the 32nd Century?

About as locked in as any of the Time Travel in the 23rd and 24th centuries.

Star Trek time travel can be inconsistent, but usually, it tends to stick with there only being one timeline that alterations shift back and forward, something that isn't really helped by the Time War.

The only time that we've seen anything approaching an alternate timeline like that is with the creation of the Kelvin timeline from the Narada incursion, which resulted in bidirectional effects that separated it into a new, independent timeline, but events like that are more the exception than the rule.

Though, normally, Trek time travel rules would suggest that anything lasting longer than a season (or into the next episode) is usually here to stay, if it's not reverted at the end of a multi-parter. Data's head remained centuries older than his body, for example, and the crew of the Bozeman are still rattling about the 24th century, having jumped 70 years into the future.

It's just a future, and can be changed. I'm not aware of any objective in-universe measure of what is and isn't the prime timeline, it's really just what the writers choose to depict as such, which events are altered by time travelers and which ones stand. Since we've seen a full two seasons in this version of the 32nd century, it's more likely that future shows will try to keep consistent with it, but it's also possible they'll be "retconned" into an "alternate reality".