Decided to answer anyway, despite my comment. Yes, you will get new growth this year, no you won't get flowers next year on your Deutzia, possibly not the year after either. Elaeagnus is not grown for its flowers, but that is also unlikely to flower next year if you proceed with your plan.
Both these shrubs sound ripe for renovation treatment, because it sounds as if they've been allowed to grow untrammelled, and have now become unproductive in parts, congested and overgrown.
For the Elaeagnus, you should really wait till winter, when the plant is more or less dormant, then saw down the branches at an angle to allow rain to run off, taking it down to around 2 feet to create a basic framework for the shrub, and at the same time removing any dead parts. In Spring, feed with a fertiliser such as Growmore, and then remove any thin and weak shoots, or shoots that you don't need or want, as the plant grows again next year. Doing this type of pruning now may force new, sappy growth which won't have time to harden off before winter sets in, and there may be excessive bleeding of sap from the cut areas, which will make the plant vulnerable to infection.
For the Deutzia, which flowers on the previous season's growth, you've got two choices - leave it as it is now, but in winter, (though you could do this now if you don't mind risking damage to new growth if winter sets in early) remove any dead parts and cut down a third of the healthy shoots to ground level, leaving the rest in place. This will mean you do get some flowers next year, and you should then prune immediately after flowering, removing any excess growth at the same time that you didn't take in winter. The other option is complete renovation - cut the whole thing down in winter, taking out all dead parts, and shape as growth appears next year, removing any unwanted growths, done around the time the plant would have finished flowering most years. You may need to repeat this the second year, for there may be no flowers then either. Thereafter, prune immediately after flowering each year.