Cutwel do an internal 60° micro threading tool that will cut 6mm internal threads. Downside is that the tool and holder cost about £100.
Alternatively, Cutwel do a 6×0.5mm spiral flute tap at a much more reasonable £9.53p, plus the dreaded VAT. A spiral flute tap should get you within 1-2 thread pitches of the bottom of the hole, which should be fine since you have fairly deep threads. The drill data for the spiral flute tap specifies a 5.5mm drill, so just use a 5.5mm slot drill to give a flat bottom to the hole.
If I was going to screw cut the threads, I'd change the pitch to give a tpi of 48, then it would be dead easy to cut on what I assume is an imperial lathe, without having weird gear trains and needing to keep the half nuts engaged.
The only way to get a full thread running the full depth of the hole is to thread mill or fabricate, ie, make a threaded tube and then screw in an end cap, secure, and machine to size.
Regards,
Andrew