Very old house needs modernising. What would you do with £2000?

Puntoboy

I agree on structured cabling, with the plethora of wireless devices, anything you can get off wireless, like Smart TVs, media streamers, etc. the better. Especially CCTV with PoE its a no brainer.

I would personally go with CAT6 at least however, especially for external runs like CCTV. I've read that it has better lightning resistance so better protection for your equipment, although not completely fallible.

maf1970

I would start with the electrics as this is a basic requirement. Go round your house with a map and make a note of all the places you have sockets and where you may want some(some rooms are usually under-provisioned). Where is the fusebox and how old is it ? Where are the meters ? Do you have gas as well ?
Second I would look at networking the whole house with it all running back to a central point(Cupboard under stairs,loft are examples). This can be done using cat 5e or 6 cabling. Personally I would recommend 6 for some future proofing and always put in more runs to each location than you will actually need then if anything goes wrong you can just switch to a spare line and fix the broken one at your leisure. It will also come in very handy for your CCTV setup.
For wifi you will need to map the house first to see if any rooms don't get wifi. This can be done by a laptop using software like InSSIDer or a mobile with a wifi analyser app on it. Things to remember are that there are two bands in use generally(2.4ghz and 5ghz).

Doorbell - 2 horse race really between Ring and Nest. Ring has multiple offerings,Nest just one. However both require subscriptions if you wish to store footage.
Ring Doorbells
Nest Doorbell
Nest vs Ring