I tend to be a "zen" about such things and mentally "write off" the entire day as having no Internet available.
I've no experience of transitioning between the two vendors in question, but I'd suggest that when coordinating activities between two independent third parties (arguably three if OpenReach are involved also) one should not expect an instantaneous transition at some prescribed time.
It's more likely that "some time" on the appointed day BT's service will stop and "some time later" Sky's will activate. Of course, you might get lucky and all the relevant engineers turn up to "do their thing" at more or less the same time in the Exchange (or remotely effect the relevant changes through a software config. change.) But I wouldn't bank on it.
Of course, there's no reason not keep using BT's service/router until it stops, then once it does plug in the Sky router and wait for it to go active.
But if I was planning this with my professional "network managers" hat on, I'd be advising the affected users pretty much as above - ie they should "plan for the worst" (no Internet all day,) but if you do find it working, you're welcome to use it, but it may go down at any time without notice, for unspecified duration - so don't plan to be doing important.