If VM simply need a local authority permit, that takes a couple of days. Obtaining a wayleave to cross land other than the public highway is a more complicated process, and can indeed take weeks like anything that involves lawyers.
VM also have been having sometimes protracted problems with new installations that have nothing to do with permits or wayleaves, with repeatedly missed construction and installation dates, sometimes taking as long as four months. More worryingly, on a small number of instances, VM prevaricate and piffle for months before deciding that the property is "not serviceable". Attempts to short circuit all that have generally been reported as unsuccessful, because VM's internal systems and bureaucracy appear designed to prevent anybody taking ownership or using their initiative. Whilst 4G dongles are often promised, it seems they are promised a lot more often than actually delivered - do not rely on promises.
My suggestion is that you sign up an Openreach ISP with a 30 day rolling contract, but don't cancel the VM installation. If VM pull a rabbit out of the hat in the next few weeks you can cancel the Openreach connection without cost under cooling off rights. If VM don't manage to give clear confidence and get you connected you can use the Openreach line for however long it takes VM. If VM decide they can't install at the property, then as you've only got 30 days notice you can then review the Openreach ISP situation to see if longer contracts make better sense.
