Vehicle recovery

[Imp]

I've never looked into the whole track repair thing, but I'd guess a 15 minute repair was a damaged track while stationary, because a moving tank leaves track spread across the countryside

As said no expert but would think need ideal conditions to repair it quickly, just think about working on something normaly.
Possible problems
As you stated probably collect track & refeed it if not its just been hit with a lot of energy as in shell mine etc, can you get to the damaged bit without driving off it.
Determine how much track is damaged & try to dismantle from there that alone could take 10mins because if try pulling a locator pin out & that section of track is twisted even minutely it aint going to come out. If any dirt sand whatever got in again taking it apart becomes not wheres the WD40 but get out the blow torch, also handy to clear any other damage from the running gear. The list of possible problems just goes on & in terms of relating to how many times have you taken on a simple 5 min job only to be swearing at it 30 minutes later. And that has not just been blown up had the shearing force of tank try & twist it or been accompanied by the sound of incoming.
In the workshop with a new track probably quite possible in the field after abuse I am guessing very doubtfull.
I would think the procedure is something like get the break to a position you can work on it possibly requiring removal of part of a side plate etc, deal with any other damage, release tension so can get it on then reset tension so it does not fall off. Sounds a lot to do in 15 mins even if everything comes apart as it should. 1 hour is more normal I would think if you want it to stay on.