Thanks again Nick W. I would need some steel strip about 5mm thick, and some steel tube 6.5mm internal diameter, and would have a better chance with some new 4mm and 6mm left hand drills Obviously you have done this job a few times before, and got the tools for the job. I still do not understand how you keep the smaller drill central to the stud. Perhaps you have a stock of jigs different internal bores.
Like welding, I am eager to learn, but lack confidence to tackle a major job like this. Some while ago I purchased a 90 degree turn drill adapter, that might come in handy on this job.
I would make the 'jig' from a solid piece of round stock about 25mm long, bore the hole in the lathe with the drillbit I intended using to drill the stud, then face off the end that goes against the head - the other end doesn't matter. Clamp both parts to something flat, and snot them together - there's no real force involved and it's a one-off job, so don't spend any time on it. As mentioned by some else, the mounting hole in the flat piece would be better slotted; you can then centre the jig over the offending stud. Doing this by eye, with a torch is plenty good enough.
There's no need for a pilot drill with this setup; go straight in with a sharp, good quality 6mm. Buy one if you have to. As I have the lathe, I would make a custom centre punch to fit the jig out of some silver steel with a point machined on the end but it isn't strictly necessary.
Left handed drillbits are good for this job, as they
sometimes unwind the stud once they start to bite which is always worth hoping for. My suggested sizes would be suitable for doing 6 or 8mm studs, not to be used in sequence.
Welding a nut on the end of the stud works well if it hasn't broken off flush. What you actually do is get a blob of weld on the stud, knock a nut over that, and firmly weld them together. Then you undo the nut
while it's still red hot. You need to be confident of this technique, as if it doesn't work you have blob of hard mig weld to contend with.