Assuming that you didn't play with the stalk, they should have started again when you drove off.
You said that they wouldn't go again when you restarted, no matter what you did with the stalk.
There could be several causes for this. It is not a simple circuit as BMW thought it would be a good idea to complicate a simple circuit by adding a wiper control module. It's at times like this that you wish you had RobertT's Frogeye
Wiper motor kapput.
Stalk controls kapput.
Wiper/Washer module kapput or things/connections that go to or from it...light switch, instrument cluster (speed output), headlight washers, wiper motor, washer pump, wash/wipe controls.
Fuse 3 kapput.
Fuse 36 kapput.
Fuse 37 kapput. Unlikely, as it also covers other things that you haven't mentioned.
Fuse 44 kapput. Unlikely, as it also covers other things that you haven't mentioned.
Earth connections to any of the components. Unlikely, as the earth connections are shared with other components.
Any of the associated wires.
If you put the build year in you profile, that can also help with diagnosis. I have guessed that it is 1998.