Okay, so I'm into the nitty gritty now, and I've hit an issue that the instructional videos and other advice here don't appear to address. I'm going to describe my set up process so far so that, hopefully, someone can spot something I've either done wrong or overlooked - or, alternatively, end up as puzzled as me.
As background, I've been tuning guitars for years and I also play piano and brass, and sing. My ear is pretty reliable (I used to be able to perfectly pitch B natural, though that seems to have gone now). Anyway, the concept of checking for intonation by playing the string at the twelfth fret to produce a note one octave above open tuning is a very straightforward one to me.
So I set up the saddles on the Tele first to eliminate buzzing and then come back as close as I dared to the buzz. In doing so I noted the neck had some back bow, so I loosened off the truss rod to try to get rid of that. Using a straight edge to check, there didn't appear to much difference. The bow didn't disappear, so I reasoned it may take a day or two to flex and I pressed on by raising the saddles to get the desired effect.
I then adjusted the saddles back or forward to set up intonation, as instructed on the videos. I'm happy with the way that's set up.
The problem is that the notes on all strings at the 1st, 2nd and 3rd frets are all a bit sharp. This effect reduces (probably logarithmically) the closer I get to the 12th. At the 1st, though it's chronic. Instead of sounding a semi-tone up, it's only a few degrees off a full tone. Not good.
To my mind this probably indicates that the strings are stretching when they're being pressed down here, producing a higher pitch. Logically, the reason for this could be that the nut is too high. However, I'm keen to hear other theories.
If it is the nut, I'd also be interested in a process to rectify this. At the moment, as you can imagine, it's not a very pleasant guitar to play down at the skinny end!