[Semaphore (OP)]

Suppose we had a program to irrigate, cultivate and hence cool the desert (OK, it would take a while...).

What would be the effect on the world's climate, specifically that of Western Europe? Would we see a long term shift in the jet stream, for example?

[Tim the Plumber]

I'm not sure that it would actually cool the desert over the course of the whole day.

Although temperatures in deserts are often very high at their peak they are cold places at night due to the lack of water vapor and plant cover to provide heat capacity to slow the cooling down.

The refective albedo of deserts is more effective at not absorbing the sun's heat than the much darker green of well watered leaves.