That's the original question: What licenses? What laws and regulations?
I spent some time searching the internet, and turned up a few things that might help.
A page specifically addressing poultry farmers and laws:
https://www.lawdonut.co.uk/business/sector-specific-law/poultry-farm-legal-issues
"If your business will prepare, store and sell food (that is, if you process your eggs or poultrymeat to add value) you will need to register with your local authority environmental health department. They will inspect your premises and help you to comply with the requirements of the Food Safety Act. You should contact your local authority early on in your planning so that you register in good time. There is no charge for registering."
https://poultrykeeper.com/general-quail/quail-eggs/
"If you are thinking of selling quail eggs, there is strangely no legislation in the U.K"
No regulation is strange, considering that there are rules for selling eggs from chickens and most other domesticated birds here:
https://www.gov.uk/guidance/eggs-trade-regulations
But sure enough, quail are not on the list.