My husbands in the military (Navy Photographer) and we use to go to the MWR (moral, welfare & Rec office at the navy base in orlando, but as it closed we didn't want to risk it this year.)
However last year we were ok at the ticket booth in Disney Land, Ca.
He just showed his ID card at the ticket desk and as a Brit you'll get it, not sure about the rest of your familty.
I have a dependants pass, which looks similar to my husbands ID so I got it as well.
As we are going to MK on the morning of our first full day, we wanted to take tickets with us in advance but the MWR office has relocated try:
www.protocoltraining.com/Seminar files/NAVY MWR TICKET PRICES.pdf
or follow the links from
www.baseops.net/marketplace/
You can also buy tickets from the Military Hotel at Disney called Shades of Green, we tried to book a room here but they are quite strict regarding FORMIL (foreign military), you can book a room only if you have a signed letter of a US base Commander stating you are on active duty in the US.
My advice is if you don't mind taking the chance and waiting till you get out there go for it...there are still plenty of discount ticket outlets if you can't get the military ones...just keep an eye on the dollar rate.
Jue
