Sky don't usually take too long. It is variable but sometimes it's as soon as a day or as long as a month.
Generally sooner rather than later
In most cases (not in yours) Sky have the link installed in advance so by the time fibre becomes active on infinity Sky also can offer it too.
The link installed links all of the green cabs in your area to Skys equipment in your local telephone exchange. So Sky fibre goes directly via Skys backhaul like their LLU service. Much better as Sky generally have much better backhaul in place and can cope with totally unlimited without slowdowns.
BT infinity has pretty heavy throttling.
I have BT infinity and Sky Fibre at home. My work provide infinity for free so I thought why the hell not... I can thus compare the two quite well.