The new timing set will certainly help, but depending on how much the machinist took off your head, even with all new timing parts, you might still find the tensioner sticks out further than it should. I've got typical head gasket thickness measurements around here someplace (not handy right now), but I'm thinking your 1mm head gasket is about .020 thinner than stock? So if the machinist took even .010 off the head, you're at a total of .030 from stock. Even with a new timing set, I'm worried your tensioner will still be out more than it should be. Maybe enough to limp you home, but still something you would need to address.
All of this is very fresh in my mind because I'm sitting here looking at my custom chain guide that I'm working on in order to bring my tensioner stickout back to acceptable range with my just completed .040 shaved head.
The .015 shims will help a whole lot, but then you run the risk of having your contact patch on the lash pads or rockers in the wrong spots. That said... I'm no engine builder, but I would be more comfortable with running with an incorrect contact patch for 60 miles than I would for that tensioner sticking out that far for 60 miles.
And as for the head gasket... Again, I'm no engine builder, but if you haven't run the engine at all, I'd be comfortable with leaving the head gasket in place if you were to pop cam tower shims in there. Follow the correct loosen and re-tighten sequence and procedure on the head bolts and I'd expect things to be OK.
Which route to take kinda depends on how quickly you need to get that thing out of your buddy's way.