Back to your original question- while you're doing the work you'll be better off installing a chunky power supply up the garden (wouldn't be a surprise if some of the larger hot tubs pull 8 or 9 kw on full load).
Talk to your electrician/certifier/building control about how deep they want the trench, what sort of backfill to use, marker tape etc etc.
To get ideas on what its going to cost you can use tlc's online calculators to give a guide to cable requirements (your electrician may or may not get a better price on the cable than you can).
Suitable junction boxes- there's all sorts, you're looking for IP66 and big enough to make terminating inside easy (SWA is fairly rigid)
You might even consider a caravan site type of box with distro in it and a 32A blue socket (for your tub) and 16A (or 13A) sockets for smaller stuff- if you're in the tub you'll want some sounds, maybe a bit of light
You may find it more cost effective to go straight from house to shed with full fat supply then use a short run of smaller gauge to the potential hot tub area
While you're trenching up the garden and to your shed, chuck a duct or 2 in for LAN, alarm wiring, anything else that might come along. Nothing stopping you putting SWA in a duct but you have to watch bend radii and stuff like that