Nope, don't do it now, wait till mid winter, as its an evergreen. Presumably you have tree like limbs on this shrub - you'll need a good pruning saw to renovate it properly. Cut down to a couple of feet at most, angling your cuts slightly on chunky limbs to allow rain to run off, till you've got a framework left at the base. In Spring, feed with a fertiliser such as Growmore, and as new growth sprouts, remove any you don't want or need.

After doing what Bamboo says, I would also take the tips off the new growth when the shoots are maybe 20cm, to help it bush out instead of just throwing water-shoots.

Thanks for the advice.
If I did cut the eleagnus down now to about a couple of feet - what do you think would happen to the shrub next spring/summer?

Exactly what would happen if you cut it in winter - except you run the risk, doing it now, that any new growth occurring this year may be killed, leading to death of the plant itself, if we have a harsh winter, and you risk infection because the newly cut wood will be very exposed and the sap is up.
I'd also add something I forgot before - Elaeagnus is not grown for its flowers, but rather for its leaves.