thanks Carpenter.
I've done some searching and it seems a box spanner or an adjustable basin wrench is what I'm after.
Thankfully, I can remove the tap completely, and then secure just the nut and the thread piece to the sink with a good tight fit, and then run the tails into the tap from below, and then turn the tap onto the threaded piece from above, much as you described. With a box spanner or adjustable basin wrench giving me a good solid fit fingers crossed the tap will be nice and secure.