Pull the cord tight and lie on your side. In really bad weather you want to be searching for some sort of shelter anyway.
But only your head needs to be covered — your feet are in a waterproof bag anyway, you're just trying to stop water coming in the opening. It's possible to knock together a tiny tarp weighing 50g or so, just to cover your head, or simply tie your waterproof jacket across your bike, laid down, and stick your head under.
I'm a bit of a Bivvy lightweight/newbie so far. Haven't used mine in the rain (although I have used it near 0'c); an Alpkit Hunka. It worked perfectly well in warm-ish Autumn Belgium and 'king freezing Spring Malverns.
SO anyway, 99% of my "knowledge" comes from
The Book Of The Bivvy, which is such a good (and funny) read that I choose to believe everything in it:
The author suggests trying to use your bivvy in conjunction with some "
shelter"; this could just be a dry stone wall. If you get downwind this will keep a LOT of water off you, especially your head. In most UK roadside situations you will be able to find even better shelter (assuming you haven't collapsed on the spot, can't ride another inch!).
Overall, his philosophy is that you MIGHT get caught in conditions where you end up wetter-n-colder than you planned, but this is so rare it is worth the gamble of taking so much less kit with you. And the next night will be better, if it's not you get a hotel where you can dry everything out