It happens to everyone, and usually at some critical point. Simply grab the tag end of the broken silk as quick as you can with one hand, letting the bobbin fall. With your other hand, reach for your superglue. You will soon master opening that with one hand. (try not to glue the tube to your lips!) Dab a tiny, barely visible amount of glue onto the thread, and wrap it once or twice.
It will stick very well. Now you can reach for the bobbin and re-start where you broke off.
(Note, some tiers dispense with the whip finish, and tie their flies off this way)
You will often find that the brush on your superglue, UV glue, or Sally Hansen’s nail varnish, is just too bulky for the fine work you do. Simply use a small pair of scissors to trim away the brush fibers, leaving a much more manageable brush size. I find I trim away three quarters of them!