If you only have the lathe then that's what you would have to use, normally it is done on a tool and cutter grinder, which is very simple and quick to do, there are many types of grinding machines that could be used also, you want to use a diamond wheel to grind the relief of the end mill
How much relief, will depend on the depth of cut you are using, so only grind enough to clear the depth of cut.
Cover the bed and slide ways, of the lathe when grinding.
These snips are just 2 machines that can grind anything you want on a cutter.