The chances of finding a place for September at a decent school are slim. Also, many of the school offices are closed or have a reduced schedule until the end of Eid.

Home schooling may be your best option for now, but as soon as you can (either while in the UK or as soon as you arrive) put them on as many waiting lists as you can. For the British curriculum DESS is by far an away the best for primary school, but it is also oversubscribed. They do follow a sibling rule, which can work to your advantage, because if a place becomes available for one of your children, the others will automatically move up the waiting lists of their respective year groups. Even so, it may take up to a school year to get all of your kids places, but if you plan on being in Qatar for a while, it is worth it.

Doha College is also worth a look, but it too is oversubscribed. There are other schools that claim a British curriculum, but I would be very wary (they are "for profit", often have inferior facilities, resources and teachers).