The ideal would be the system I helped to set up at Cambridge Woodwind Makers for Daniel Bangham. This had a Waco vertical milling head mounted on a rotary table (horizontal axis) to drill at any angle (45 deg only is limiting you to a particular instrument), aligned with a manually rotating pair of centres to hold the spindle. The angle of the spindle itself was measured with a Wixey encoder, quite cheap.
That is serious £, but the Sorby jig or a home built imitation would do a good job. You will need to be able to lock and preferably index the angle of the spindle in the lathe, and incorporate a protractor to set the angle of the drill guide. Then a means of fixing it in the banjo, but you must develop a way of setting the banjo to better than 0.5 mm along the axis. Yes you do need that accuracy for a woodwind. Though you can compensate for tuning by altering the hole size, you then lose control over your venting. I suggest you get pointed rods made up for your drill sizes, so that you can see accurately where the drill will locate by putting them in the drill guides first and seeing where the centre is. It is very hard to judge distance on an angled approach.
Drilling by hand is fast enough and far less scary
. No need to use an end mill, spur drills work fine if the guide goes fairly close to the spindle.
MikeG is right for the occasional hole, it is easier to drill it on a flat. But this is not right for a woodwind for two reasons. First, you may want several holes at different angles around the axis. Second, even a gun drill doesn't drill as straight as the straightness of a turned spindle, and you won't know accurately enough where the centre of the tube is. If you put the hole in first you can turn the outside to be really concentric.
The best technique of drilling a long straight hole for a woodwind (without a gun drill) is to use the lathe. Hold the headstock end firmly (ie with a chuck, not with a Steb centre) and the other end in a fixed steady. The eighteenth-century makers just used a hole the right diameter in a fixed wooden block, Then start the hole with a gouge, and continue all the way with a shell auger. The straightness comes from the work rotating and the drill staying fixed (apart from translating) as this tends to centre the drill. You can then enlarge the hole to a precise diameter with a D-bit which you can grind yourself, and shape it if necessary (eg for a bagpipe chanter) with a reamer. Making the reamer is the trickiest part, but that is another topic.