Interesting slice of real life but probably not good PR for farming; as there was a good example of how NOT to lamb a ewe, the lamb being castrated was probably over a week old and the sawing off of the sheeps horns should only be done by a vet (as far as I'm aware). The children were the highlight of the programme for me.
There's even more of that lambing in the longer version, and BH & I were both of the opinion that we'd have had the vet to that ewe - or taken her to the vet - some hours earlier. I'm sure lambings like that are rare in pure Swale lambings; I lambed 450 Swales for a few years, and bad lambings were very rare, particularly with the ones that had gone to the Swale tup. (As opposed to the BFL for Mule lambs.)
Which said, I wish he'd used more lube.

I don't know that I could guess the age of the lamb being castrated. I imagine they keep the ewes and lambs close by until they're tagged and ringed, so they can be sure they get the parentage logged correctly, and I would think you wouldn't keep them hanging around the farmstead longer than essential - so probably a few days, not longer than a week.
Sawing off the horn where you do not need to go into the blood vessels (and hence nerves) can certainly be done on farm. It's routine, like trimming feet. We took a Swale tup to the vet to have his horn trimmed because it was certainly going to need to be higher up. The vet showed me how it was done in case we had another.