As you're not in the States, you won't have a 'phytosanitary seller'.
This is commonly known as brown rot in the UK, and is pretty common - sets in usually as a secondary infection following damage to the fruits, usually from plum moth or birds pecking at them. Select one of the less affected fruits, cut it in half and see if you can find a sort of tunnel with bits of brown excrement, and if you're really lucky, you'll see a little pink larvae in there, towards the centre. If you do find that, its plum moth, and you need to use a pheromone trap for next year, and spray at certain times. The RHS has a page on plum moth which tells you when to spray and what with.
Other causes of brown rot are too many fruits all touching one another in damp weather, but unless you're in a wet region, this year, that won't be the cause because of the hot weather.
UPDATE IN RESPONSE TO COMMENTS:
No criticism of your language intended ondoteam, I thought you were speaking about some local authority to contact - I understand now you are talking about using a fungicidal treatment/spray.
Duncan Morris; to reiterate, this infection is common, and there is no treatment once it's taken hold. To avoid this problem in future years, establish what has caused the original damage to the fruits to allow this secondary infection to set in - as this summer has been hot and dry, the cause won't be damp weather, which may, in some years, cause this trouble, but rarely. My money's on Plum Moth.
UPDATED ANSWER 6/8/14 - You've confirmed you think Plum Moth is an issue - you'll need to take action next year or you'll end up with no plums again. Get a pheremone trap appropriate for Plum Moth, hang it in the tree at the indicated time (see packaging). You'll also need to spray (Bayer Sprayday greenfly Killer or Westland Plant Rescue Fruit and Veg Bugkiller) beginning to end of June I think, check instructions on the pack. The pheremone trap might be enough to protect most of the fruits if you have only one plum tree.
UPDATE 2 15th August: I appear to have been downvoted on this answer, but given the answer is entirely accurate, I'm somewhat puzzled as to why, so will whoever marked it down please give an explanation?