Will the flow be significantly enhanced by 22mm pipe even though
the upstream pipework is 15mm?

It all depends on the length of pipework. For a simple shower in a normal
house with standard layout, I suspect 15mm will normally be fine. However,
there are circumstances in which 22mm would be better, such as long runs,
panel showers etc.

15mm has some advantages too. As well as being easier to
install/route/conceal, it means that the hot water becomes available sooner
if the pipes are cold.

Christian.