skyscraper101
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Yup, this type of stuff takes months, funny poster above should mention Liverpool, because at my old work place we had to wait 3 plus months to get a line into a building that was already full and required digging up the roads etc, this was done with Openreach thou.
Councils take ages to authorise it.
As the BT line could take as long as Virgin to sort, you 'could' try playing devils advocate and get both installed. If (big if, I'd expect the Virgin one to be honest to fall even further back) Virgin install within the 4 weeks quoted and BT the 3, you could, cancel BT within the cooling off period without getting an additional charge. Risky thou, as I know if it was me, this just wouldn't go to plan.
Could you not ask a neighbour if they wouldn't mind sharing their Wifi for the time being and offer them a cash incentive as well as showing all your letters etc to prove you're not pulling their leg?
For the most part I've just been tethering to my phone. But this is shit because I want to connect my echo dot, and smart TV, and various other devices and I'd burn through my tethering allowance in no time if I started with all that.
The neighbours WIFI is too weak sadly anyway.
Do BT have a cooling off period? I think I'd get lumbered with having to pay for the install cost if I cancelled anyway (they need to install a fibre line, I checked).