No - it's always the man's name (both first and last)which is used ( if the couple have married and the woman has taken the man's name) and his title always comes first too. So it would Mr and Corporal John Smith.
What I would say is that a military title is not the same as a professional title in that doctors and dentists etc will use their title of Dr in their everyday lives.
My experience if the military is that they don't. So a general letter from the bank wouldn't tend to be addressed to Corporal John Smith rather Mr John Smith
My DH is a dentist so his title is Dr. His sister is also a dentist (so has the title Dr) but she's an army dentist too so also had the title of major.
When we invited her to our wedding, her invite was addressed to Dr x x not Major as it wasn't an army affair. However, when she has previously been invited to military weddings by colleagues, her invitation is addressed to Major x x
So my long winded point is that if you and/or your DP are in the military which I presume you are! and there are lots if military types then I would use the military titles but, if not, I would consider just using "mr" etc
Sorry - should have made that clear in the first post