Welcome to this user forum.
Do you have normal ADSL broadband or superfast "fibre" broadband?
Do the lights on the BT Home hub change colour when you lose your Internet connection?
I'm confused as to why the internet going down stops you watching any TV and you mention boosters and that would normally refer to boosting a normal over the air TV signal, nothing to do with the internet.
Sick to the back teeth. BT are a disgrace. My Internet completely disappears between the hours of 1am-6am every night making watching any television impossible during the night. Had problems since moving into property 1 year ago. Despite numerous attempts BT have never sorted the problem. Kept promising to send me top of range boosters but only ever sent me the cheapest version.
If there was another provider in my area then BT would not be given another penny from us.
A complete disgrace
If you've not set any form of access control between those hours, then as you've implied it has happened for a year since moving - to me that could indicate some form of interference. This is assuming that the problem only appears on wireless - is a wired connection OK?
Do you know of anything that might cause interference, machinery that works at night - not necessarily yours, but a close neighbour etc.
If you are referring to normal TV then my question would be do you have a Youview box or similar recording device and the aerial feed is via that unit?
If so then it's likely the box is going into sleep mode & not passing the TV signal through to the TV.