There are several organic remedies you can use:

Spray with dormant oil in the winter, before the leaves open, to kill aphid eggs. They will still fly in from your neighbors, but that will slow the infestation down.
When the leaves first open, start spraying regularly with neem oil or cedar oil. That will repel flying blackfly, and keep them from getting started. Once colonies are established, these two are slower, and less effective.
Commercial insectaries sell parasitoid wasps. These are tiny stingless wasps that lay their eggs in blackflies. The larva kills the blackfly as it feeds inside it, turns to an adult, breaks out, and lays eggs in dozens more blackflies. After several weeks, they can bring the blackfly population down to a few clusters. You may have to search a little to find the most effective species for your kind of blackfly.
Soap and organic washing up liquid can be effective, but they must be applied thoroughly and frequently. They only kill the bugs they actually touch, so any that you miss can repopulate the plant within a week or two!

Good luck!

Blackfly do love these shrubs, that's for sure - spray for them, but make sure you use a systemic insecticide, not a contact one - these will keep the blighters away for up to a fortnight, because the insecticide gets into the sap stream of the plant. Also use correct spraying technique - spray the underside and topside of the leaves and young stems till run off - that means the stuff's dripping off the leaves. Don't spray open flowers, but you can spray flower buds. You can buy ready made sprays, but it's always best to get yourself a decent sprayer with a trigger, and buy an appropriate insecticide in a bottle to mix up the correct solution yourself. Having a trigger means you're not constantly squeezing the sort of trigger you get on shop bought sprays, you just thumb it on and then off when you're finished - check out Polyspray's ones if you want to buy a sprayer.

How about underplanting with marigolds or nasturtians both of which are loved by blackfly, they might like these instead of your mock orange who knows, good luck.

I blinked when I read Grandmage's recommendation. Marigolds to attract blackfly?! Then I realised that she meant English Marigold (Calendula), instead of American Marigold (Tagetes). The latter will repel blackfly, but that only helps in warm weather, and only to about 30-50 cm away from each plant. I didn't recommend systemic insecticides to Lyn because she seemed to be looking for an organic solution, but some form of Provado sounds like the most effective solution, as long as she isn't planting something edible within a couple of meters of the edge of the bush.