It seems to me that at least some of those answering here do not grasp what is at issue, I hope I do. Firstly, for a non-UK citizen to hold and renew a driver's licence he/she does not need to be English (can be Welsh, Scottish or even foreign) and this has absolutely nothing to do with his/her partner's/spouse's nationality (which equally may be English, Welsh, Scottish, foreign). The UK authorities restrict issue of licences to people whom they can identify from/by some documents from a limited/strict list of documents/proof of identity and the same applies to renewals.
In your case I suspect that the quickest, easiest and possibly the only way is for your partner to obtain and present a passport from her country, this presumably being one issued by Eire/The Republic of Ireland. I suggest she applies for one without delay and obtains it in the quickest possible way. She should then promptly submit it with her licence application by registered post and I would recommend enclosing a registered self-addressed envelope and requesting it be used to return the passport.