I live in London and have no car. Been up to Skye a few times in past decade, but May and September.
First time took sleeper to Glasgow, wonderful scenic train journey to Mallaig. Ferry to Armadale was late and missed connecting bus towards Portree. Good prices booking train in advance.
Without car best and cheapest way is probably overnight coach to Glasgow (presumably one goes from Birmingham) , then Citylink bus to Skye, 5/6 hours (scenery good). This has the huge advantage of going directly through Broadford with a supermarket (road leads from here to Elgol, buses) and Sligachan which is probably best bet if wanting to climb next day (but this and all campsites prob now closed) which has expensive hotel and also bunkhouse (may be open all year), no shop.
The second major climbing base, Glenbrittle, is 7 miles down a road with no buses (roadhead reached by buses from Portree). Think hostel now closed but has BMC hut. No shop this time of year.
Definitely download local bus timetable to Skye. Fares good, often school timed buses. Found hitching on minor roads easy, but not on major road to Portree (but more buses). If using tent wild camping prob necessary. There are no regular ferries this time of year from Elgol to Loch Coruisk however bus or hitch to good point for climbing Blaven.
Prob hostels closed November but imagine BnBs have plenty of rooms at reasonable prices outside tourist season. Camasunary bay approached from Elgol or Elgol road had a very good bothy (now moved buildings but still apparently OK, check MBA site).
Another travel possibility is plane to Glasgow (buses to Skye go through here).