I think the biggest cost for the sparky will be time to route the cable, personally I would happily take the hours/days it would take me to route the cable for them to turn up and connect at both ends but I’m sure with all the part p and refs etc they have to do all that as well....
No, it's absolutely possible and sensible for you (or your gardener or other labour cheaper than an electrician) to do the digging/drilling work and have the electrician just do the actual electrical work. Most electricians don't enjoy digging and will be happy to quote you a cheaper price on that basis. However, it's critical that you contact the electrician and agree what's to be done BEFORE you start, rather than presenting them with a buried cable and saying "use that!". You are asking them to put their signature to the "design" section as well as the "erection" and "testing" on the certificate, so they need to design the work. They might want to see the cable in your trench before you fill it back in again, and check that you have put warning tape in the trench etc.
It is also theoretically possible to do it as a pure DIY job, with either local authority building control or a 'third party verifier' to inspect the work and satisfy the requirements under Part P, but for a modest job like this it's normally more expensive than getting an electrician to take responsibility for the whole job (LABC or the 3PV will normally need at least two visits to inspect your first fix and to test the final result).
I suppose the sparky can run a regular cable inside the house, and then junction-box for an armoured cable outside ... but, yeah, its carrying quite a lot of umph, so reasonably thick. I'll bet that
@arg would know
It might be that through the house is sufficiently shorter that the through-the-roof cable would be even fatter? (more cost, maybe more effort to "handle", particularly as a longer reel will be needed
You are talking at least 6mm² cable, possibly 10mm² depending how long it is and whether you are just putting the chargepoint on the end, or plan to put sockets/lighting in your garage as part of the job.
Armoured cable at those sizes is 15.4mm/17.9mm diameter. You can indeed use a lighter cable such as T&E internally and then junction to the SWA where it gets buried, but non-armoured cable buried in plaster will need protection with an RCD at the source end (normally with a circuit like this the preference would be to put the RCD at the garage end).
I don't much like the sound of your lighting cables notched into the plasterboard of the ceiling, though I can't find a specific regulation that they break (except perhaps the collapse in the case of fire: did you use metal clips to retain them?). However, that's probably not a feasible way to route a 6mm² cable regardless.
As regards the overall job, if you are going to this much trouble, then I'd be strongly inclined to finish the job and get your neighbour's permission to continue the trench under his garden entrance and into the garage block so you aren't having trailing cables across the common walkway. You could even offer to let him drop his own cable into your trench in case he wants lighting in his garage (if he's not sold on the idea of EVs).
