Hi guys
Just had a very similar issue on a 2003 2.5i Z4 - headlight washers popped up when activated, one would have a TRICKLE of water come out, and the other nothing!!!
First of all tried clearing out the two filters on the bottom of the washer tank in the engine bay just in case - tiny bit of gunk but generally ok. Did not fix it.
So, bumper off (actually quite simple - don't panic for DIY!!). Drained washer fluid (again!), popped off the pipes and removed the two headlight pump units from the inside of the bumper...
Do do this push the pop-up part carefully from underneath until you can get two fingers under it from the top side to hold it in place. Then carefully remove the covers/caps from the top. Look carefully and you will see they are held on with a pin at each side. One is locked in place, and the other has a slit to allow it to 'twist' off. Twist it off carefully whilst not twisting the head of the washer pump.
Then two plastic lugs on the pumps pushed upwards allow them to slide out of the bumper.
I cut the bottom 1/3 off of a 2 litre bottle or water, and put a good amount of degreaser in. Then submerged the heads and left for ten minutes.
Then the same in another bottle cut-off with boiling water for a few minutes.
Aggitate them a bit by stirring, whipping them in each solution rather than just leave the steady for the full term as this will help dislodge any gunk and allow the fluids to get in deeper.
Then, remove the green rubber o-rings (they can deform in degreaser/boiling water!), and turn them the other way up (spray nozzles at top) and submerge the lower half into the degreaser and pull the top half up, which will draw liquid into the body of the pump via the hold you just removed the o-ring from. Do this a few times - be sure to release slowly as they hold a lot of pressure!!!
Then do the same in the boiling (by now just very hot) water a few times.
By doing this you are cleaning the nozzles from the outside and inside.
I was shocked by how much discolouration I got in the degreaser (it goes creamy white) from gunk on the INSIDE of the pumps.
This gives them a damned good clean out.
Put o-rings back on, snap bank into bumper, re-connect the pipe and test before putting the bumper back on.
They will take 6-7 tries to get the water pressure built up again (I had to give mine a helping hand by pulling one up when it made the right noise - just the once to get them going!) so don't dispair if they don't work immediately.
After a few minutes testing you should get full-on spray power back!
If not, bad luck. They're probably beyond repair and you'll need to get new ones.
But hopefully this process will fix both for you, or bad case scenario fix one, or worst case fix none as they're really shot.
But worked for me and went to virtually zero to fully operational!
Hope that's of some help!