Are tube drivers allowed to have a non-LUL pal in the cab with them and
let them press the buttons that make the train go?

I presume not, though I saw it happen yesterday - it was a really old
guy, maybe the driver's father-in-law or something), and as I was
sitting in the front carriage, I could hear things like "oooh!" and
"you've just made the train go!".

I'm not going to say where this happened, as the old guy was obviously
totally ecstatic to be allowed to do it - but I just presume it's not
exactly a regular occurrence?!