The is quite possibly a delay between the BT system knowing “we” are now live, and the system your current provider (Sky) uses. It could be a day or two delay, it could be longer – hopefully others will chime in.
It could be considerably longer for Sky than most other providers.
All ISPs using the BT Openreach FTTx system use the same equipment from the end user to the parent exchange (in most cases the same exchange your voice service comes from, but not always). Assuming we're talking about FTTC, you'll have the same:
* BT provided VDSL2 modem
* faceplate filter on an NTE5 'linebox' master socket
* metallic pair (copper or, in some cases, aluminium) to the PCP (green cabinet)
* tie cable pair from to the DSLAM port in the FTTC cabinet (new green cabinet)
* fibre link from the DSLAM to the exchange FTTx switch for broadband
* tie cable pair from the splitter in the DSLAM cabinet to an E side cable pair in the PCP for voice service from the exchange
Depending on the ISP, either you or the ISP provide the router that connects to the BT modem.
From the exchange to the ISP, most ISPs use the BT Wholesale network. This seems to be provisioned from the day the exchange goes live.
TalkTalk and Sky do not use BT Wholesale. Both these ISPs connect to the BT Openreach equipment at each exchange where they offer FTTx service and use their own network for backhaul. There is often a delay between the exchange going live and this connection being made - it can be several months, maybe even longer. Ultimately, it is up to TalkTalk and Sky which exchanges they cover and when.
The link to the BT Openreach equipment is the easy part - the difficult part for Sky and TalkTalk is upgrading their network at each covered exchange for FTTx traffic.