My advice would be to get your applications in to EVERY school that you like the look of.  Just e-mailing them to enquire about places is not enough, you must get your applications in, including all the necessary documentation.  Then KEEP phoning them every week (seriously!) to find out what the situation is with regard to available spaces.  That way the admissions officer at the school will definitely remember you.  I managed to build up a very friendly relationship with a couple of schools this way.  Also, try and get the kids out here for a holiday before you move out this will definitely pay dividends - you can do the rounds visiting all the schools and in turn the schools admissions officers can meet you and the kids.  You can also let the kids do the entrance tests at the schools (not nice I know, but a necessity out here).  I dont know if the British passport angle will work but theres no harm in trying.  Also keep in mind that there will be lots of kids leaving Doha in the summer who wont return due to parents changing jobs etc (it seems to be a natural time to move job, country, employer etc) so places may well come up that way.