Go to your Welfare Office. Ask the Unit Welfare Officer to contact SO2 G1 Community Support at HQ BFG, BFPO 140. They will advise. Normally if your passport runs out of date whilst overseas the MoD will pay but I'm not sure about them paying for a name change.
Failing that ask the clerks in the Regimental Admin Office (normally close to were you get Tax Free forms, Fuel Coupons etc).
The online/Passport Office forms don't know anything about BFG or squaddies in general.
If the MoD will pay, you might have to pay up front and make a claim back via JPA later.
My Mrs did not change her passport until it ran out as its cheaper, when you book travel etc just do it in yr maiden name, and have a marraige cert (copy) in it, she traveedl for over 5 years like that no problem what so ever with customs or border control in any country (Malta. Tunisia, States etc)
Discovery001's is right. You only have to change your passport if you change your name, so don't change it until the passport runs out (or ever). Remember to book your air tickets in your maiden name so that they match the passport. I don't think that airlines and passport control people will even want to see a marriage certificate.
Unless, of course, you are a serving soldier yourself, in which case a name change may be obligatory.
Do it in UK. The cost for a name change is the same as a new passport although you do get the years left credited to you. Just be careful that, if you are on air trooping with your missus, it goes in her maiden name. (That's if families use air trooping anymore?)