Caecilius wrote:

1. TV

Standard 75 ohm coax from the loft to a socket in the living room.
I've not used TV for ages, so I assume the new digital TV still uses
the same cable

I'd put in at least three coax cables, use foil and braid screened coax
cable

and connectors as the old analog TV from 20+ years ago.
Or have they finally replaced those horrible belling-lee connectors
with something better?

F-Connectors

2. Network

CAT 5e from RJ45 sockets bedrooms and living room to a multi-way
socket in one of the bedrooms.

Guessing where people will want the sockets is the tricky bit, most
people probably just use WiFi for PCs/Laptops within the house, put a
couple with the TV aerials or internet connected TVs and games consoles

3. Telephone

Might as well use CAT 5e for this as well I guess, to avoid getting a
seperate reel of telephone grade cable.

yep.

Can I install a BT master socket and just leave it to BT to connect
the A/B pair, or are only BT allowed to do that?

Openreach let builders do that, talk to them they'll supply cable and
conduit

Is it actually worth installing telephone, or does everyone just use
mobiles or VOIP now?

Most people will want a phone line if only for broadband.