Right, flameproof overalls on; here are my thoughts!
If runout of the hole is critical I wouldn’t use the lathe. I’d drill the hole in the vertical mill and then machine the rest of the part using said hole as the reference for set up. The problem with using the lathe is that it relies on the drill being perfectly aligned axially and dead parallel to the rotation axis of the headstock. The one thing you can gaurantee is that it won’t be.
I agree with Ramon that the 5.5mm drill is too large, the reamer will be rubbing or not even cutting. I wouldn’t bother with a pilot size drill, just go straight through with 5.3mm. Larger drills will tend to follow any ‘wander’ of smaller drills so why not start with the largest, and stiffest, drill. Less ‘pecking’ needed too. Also, keep the feedrate up, that minimises any errors due to slight asymmetries in the chip size.
I also think that 400rpm is a bit slow; I’d be up at about 2000rpm for drilling, and half the speed, twice the feed for reaming.
My sequence would be; start the hole with a carbide spot drill, drill through with 5.3mm, ream 7/32″.
Here’s a thought, you could consider a carbide drill; like for like carbide is much stiffer than HSS.
Regards,
Andrew