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Big thanks @Airhead, this is what housemate came back with:
Two coats of Zinser 123 bullseye primer. Applied with a brush. We did exactly the same on the test doors and they were fine. Current guess is maybe when we wrapped the brush with clingfilm between coats. Perhaps clingfilm has some coating that is reacting with the paint??
Mishk-
Zinser 123 is a good primer. I usually use an undercoat on top of that from the manufacturer of the paint. Zinser is very grabby which is why it's a good primer for any surface, but when you are painting on top of it it can dry the paint too quickly so when you touch over the paint to blend it the freshly applied paint becomes stringy and doesn't dry to an even eggshell. It's hard to see from your photo what's going on exactly but it could be that.
Certainly when you're painting flat surfaces you have to work very quickly and avoid going back over your work, difficult on shaker door panels like you have there.
I've seen kitchen painting projects in water based paints abandoned because of the issues. There is a great new paint base called porcelin which a Fulham decorators suppliers helped develop. It dries very slowly so that the surface can form like an oil based eggshell and it's very hard when dry so it doesn't wear like water based on shelves etc. Only downside is the re-coating time is 8-12 hours, and it costs £35 a litre or something like that.
Airhead
Looks like water based paint. Kitchens are greasy places so you have to be careful with prep. It's why a lot of kitchen painters stick with oil based paint. You can contaminate sand paper with grease so that it just embeds it in the surface. Also check the quality of the primer (it is primed right?), I usually use 2 coats on water based kitchen jobs.
Then you can get very ugly results from water based paint just from the application on flat surfaces so it's best to use a top quality paint base like little green, which is a 3rd generation water based paint and apply it with a brush that's engineered for the job, something like a wooster alpha. In any case, even with perfect conditions, tool, paint and lots of experience it's a pain.
You should probably clean all used kitchen doors with meths or white spirit at the start of the prep.