So I applied to see if the children could get emergency travel documents, and I guess the restrictions on their father's entry visa didn't qualify. We leave a few weeks and I am not sure what my best approach is here. I really want us to all be able to travel together, and I don't want the children to miss the start of the new school year.
Yeah, unfortunately it's not possible to expedite applications if it's their first passport.
First UK passport applications from the overseas take around 10 weeks for processing, possibly up to 16 weeks.
In order to apply for their UK passports, I believe you will need to send their US passports to the UK Passport Office, so your children will not be able to fly to the UK while the applications are processing.
I suppose you have a few different options:
1) Leave the kids with family or friends for a few days while you and your husband fly to the UK to activate his visa and pick up his BRP card (this will start his official UK residency). Then you can fly back to the US to apply for the kids' UK passports, and fly back to the UK again once they have them.
2) Fly to the UK to activate your husband's visa and pick up his BRP card, with the kids entering as visitors. Then fly back to the US with the kids to apply for their UK passports. HOWEVER, your kids may not be allowed into the UK as visitors if you are entering on a British passport and your husband is entering on a spousal visa, as you will be considered to be moving to the UK (not visiting), and UKVI may not believe they are genuine visitors and will leave the UK again... so you may end up right back on a plane to the US.
3) You could try to enter the UK without the kids' UK passports, but with proof of their UK citizenship (their birth certificates, your birth certificate, your parents' birth certificates, your UK passport etc.) and see if UKVI will let them in as British citizens... then you could apply for their UK passports from within the UK. I'm not sure if they will be allowed in or not though.
4) Delay your move to the UK for 3-4 months and apply for the kids' UK passports now. In order to be able to enter the UK after September, your husband would need to apply for a new replacement 90-day visa vignette (£154) so he can travel after his current one has expired. See here:
https://www.gov.uk/transfer-visaOptions 1 and 4 are probably the safest options, as in those cases your children would not be attempting to enter the UK without UK passports.