Melamine paint might stick to it but since it is outside I would not hold my breath and melamine paint chips easily. It is relatively expensive. You should be able to get it in a rainbow of factory colors at a real paint store and some companies now carry it in tintable bases like for other paints. I haven't heard of it used much for exterior applications but it is gaining in popularity for painting over melamine and plastic laminate cabinets and so forth. It is sticky stuff and rather hard to work with.
I think a better option would be to use a plastic primer or plastic paint. I am not sure what you can get on the other side of the pond but the best choices here are from Rustoleum and Krylon.
Rustoleum makes a plastic primer that comes in rattle cans or liquid form. You can paint over it with anything. The company also makes self-priming plastic paint that comes in rattle cans but the product line has a limited number of basic colors. I have not used the Rustoleum products but the company has a good reputation and does not seem to make ridiculous product claims.
Krylon Fusion is another self-priming plastic paint (great for other surfaces too) and comes only in rattle cans but in lots of colors (not sure the following color chart is current but you get the idea). I have used it with great success on plastic garden furniture and even kids playsets that had faded. Obviously your project will take a few cans.
You will need to clean the surface thoroughly and scruff it up a bit with fine grit wet/dry paper or steel wool to improve adhesion. I would clean up the metal parts that are bleeding and spot prime them with a rust inhibiting primer. With rattle cans remember multiple thin coats rather than trying to cover in one or two is the way to go.
If you cannot get plastic paint or primer, a water-based superbonding primer followed by a couple coats of acrylic house paint may hold you for awhile.
Good luck! I think it is a rather cool looking find and can see it will make a great shed! If you think of it, post an after picture.