I have asked a moderator to provide assistance, they will post an invite on this thread.
They are the only BT employees on this forum, and are based in the UK.
Bear in mind that broadband is self install anyway, so a visit is not made. The service could well have been activated, and the phone was working, then the BT home hub would have worked unless there was a fault.
The fact that your broadband had been activated, and you then asked Sky to provide service, would mean that you broke your contract with BT, hence the cancellation charges.
The correct way would have been to cancel during the "cooling off period", you would have neede to make it clear to BT that is what you wanted to do, return any equipment, and wait for confirmation. You would still have to pay phone cancellation charges if the phone had been activated beforehand.
Then you would be free to place an order with another ISP.
As it is, the only way I can see to avoid these charges, would be to cancel the Sky order, and stop the cancellation of your BT Broadband service. If that is what you want to do, then one of the moderators may be able to help.
Ports on the fibre cabinet are available to all providers.
All I can suggest is to see if one of the moderators have any suggestions, but I suspect that you may have to stick with BT to avoid those charges. There may well be a spare fibre connection available, it could simply be an error on the Openreach database which can be corrected.