UKVI make it so difficult.
I think we have also seen a few cases where the British spouse returns to the UK with the children so they are able to get settled into schools (if the children are British and don't also need visas).
Planning temporary accommodation on a month to month lease is probably your safest plan. You may be able to find some in the same school district you're already in? It's not at all easy, but know you're not alone!
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My British spouse is willing to bring them over ahead of time, but we will start out living with his family briefly and not sure if the school in that neighborhood would have a "place" for our kids anyway. (We definitely don't want to stay there.) So it's the same problem, in a way.
Does "Priority" really speed things up significantly? How fast is the turnaround, generally? (I know it varies by time of year.) Even if they did go ahead of me, I wouldn't want to be away from them for too long. I can't imagine being thousands of miles away from my young kids and husband for weeks on end, or longer? I would be freaking out I would get denied, and what then? Terrifying.