I have a few suggestions/comments. I know nothing about your side of the world, but I have played in the Caribbean. You seem to have a couple of problems/decisions.
1. Your husband will need something to do while you & your children are diving. Topside activities may play a role.
2. I believe your biggest problem involves flight and lodging, not as much finding a dive operation to take care of you. Referrals in scuba is an everyday event in vacation destinations. Some locations are more impacted than others, but generally, there will be room for a few more customers.
3. As others have said, you may wish to have your children start the training in Jersey for the following reasons:
a. Pool and class time won't cut into vacation time.
b. It will give you an opportunity to get a refresher.
c. It gives an opportunity to select mask, fins, boots, and snorkel that fits, can try in the pool and exchange if necessary before the trip.
4. Be aware of the cruise ship schedule. You can either be far away from the crowds or you can go down to that area if you need to shop for souvenirs. For example, when the cruise ship is at Grand Cayman, I go to the east end, or on St. Kitts, I dove in Nevis. You may have other preferences.
5. Azores will be about 20C to 23C water temperature, the Caribbean will be around 28C. The difference between wetsuit and lycra suit.
I took a quick glance at air fares from the London area to our side of the world and the prices were similar going to Bermuda, Bahamas, Grand Cayman, Curacao and St. Kitts. You have a lot to choose from. The pricing on scuba is pretty similar in the region also. Some islands cater to a higher income crowd (Bermuda, BVI) but there are values to be had. You will pay more for lodging and meals depending on the island, so if cost is a determining factor, location plays a role and there are some ways to keep a budget.
I hope you have a fantastic vacation.