I got an EHIC with DH (I was a USC, DH a UKC) a few years ago and there was no problem getting the card as a non-UKC.
The only thing to watch out for is that the UKC is listed first on the application (it asks their citizenship, or at least used to) and you (a non-UKC) are listed as one of the others on the app. You'll still get your own card and everything.
I don't remember the details as it's been a few years, but basically, if you're eligible for the NHS, you're eligible for the EHIC.
I can't remember how long it took to get the cards, but if you're travelling in August, I'd apply asap. The application is simple (especially after visa applications) but you do need to wait for the cards to arrive.
We've only had to use our EHICs once and it couldn't have be easier. They photocopied the card and charged us the normal price that residents of that country would pay. That was it. It was almost too easy. We were sure to get a receipt just in case we needed it or our home doctor wanted to know what was done.
If you do need to use your EHIC, be sure you take the card with you to your appointment and go to a provider who's the equivalent of an NHS doctor (not private) and remember the EHIC doesn't replace general travel insurance.
Have a great honeymoon! Where are you going?