Yes, you can transfer your service to a new home. You will also be able to keep your number since it is on the same Exchange.
The fees involved are as noted in the
legal bit:
If you want to carry on with the contract you've got at the moment, you'll need to pay £65 to move your broadband. If you need to have a phone line put in, it'll cost £49.99.
You can find out more information
here. You should contact the number provided on the website as soon as possible when you know it is confirmed to prevent downtime. Ideally, as it says on the website, with 2-3 weeks notice.
By the way, those are some nice G.fast estimates. Mine used to look like that before the service was live if I recall correctly.