Tungsten isn't a very easy material to turn I believe and will require a good lathe with correct tooling/inserts. I think most tungsten is ground rather than turned.? But I'm not a lathe guy so could be wrong.!

But if you are wanting something more than a hobby machine I do have the perfect machine that will do this standing on its head.
It's a BIG Tarex Burger twin ATC turret lathe with hydraulic Collet chuck, Fanuc 11T controller, and Fanuc AC Red top servos, it's a serious bit of kit that is the size of a small car but it's your for £2k.