I've previously screen printed over watercolors but would be interested in looking into digital printing onto paper thats been painted...
Is there any common process for doing this or is it just not done?
There's no technical reason you couldn't load a painted sheet in any digital press.
As you probably know, you have two choices:
Laser could be sketchy. You may have trouble finding a shop that wants to be your guinea pig. Due to the heating requirements of the laser process, things can easily be transferred from the substrate to the machine. Depending on your watercolor technique, that could definitely be an issue.
Ink jet should be painless. A little more expensive, but painless. There are plenty of print-on-anything ink jet printers in use today. I would test a few paper types for bleed and general printing quality, but it shouldn't pose any problems.
My apologies but I disagree with plainclothes on his comment of "There's no technical reason you couldn't load a painted sheet in any digital press". There are issues that are in place and if you do digital printing depending on the medium, ink, clear, or any chemical facet used to make that watercolor can have a bad reaction and become a nightmare or destroy the watercolor.
The reason why I asked what medium was used because depending on the substrate, in this case plastic, it can hold up to a flat bed printer. Some flatbed printers even allow for a roll-roll print (see link video). However, from the heating process if the medium is heat sensitive it can melt the material and destroy the watercolor and/or running through a roll-roll flatbed printer X amount of length is needed to tighten for the print.
Furthermore to flatbed printing the solvent inks can have a bad reaction to the material or ink from the watercolor. Another option to flatbed printing is with something they use or is labeled as Direct To Garment printing. Even then when printing apparel for garment printing if you use something other than 100% cotton it has to be pre-treated. Can this pre-treatment destroy the watercolor work? YES. Some heads can be configured to print on certain mediums where the flat material is placed and the bed or head move and it doesnt use a roll, Example video.
Roll-roll digital printing may be practical if this is on a roll. Again, if the medium the watercolor is something as thick as felt I dont know any shop that would risk running felt through there 60K digital printer.
My advise personally, and this is depending on the size, medium, and where it is going. I would possibly look into screen printing. It would be a better cost effective way, but the OP still hasn't edited his question so I am answering this on what I've seen.
NOTE: Be advised. I do not know or will bet any shop will guarantee the work and will more than likely tell you they want payment up front to cover their time and ink costs.