We have traveled extensively in the British Isles several times and visited all the countries there. We love them all.
We have found that it is best not to plan traveling over long distance by car. The farther you have to travel from point to point the more time you waste traveling (not touring). Also, you spend more $$$ on travel. This would be especially true for a visit to Normandy.
Our last trip to Britain was only in England and South Wales. We spend FOUR WEEKS having a rental car and it was amazing. We didn't do London again.
You only have two weeks and you plan to do London. I suspect you will want to do at least 4-5 days there, leaving 9 to 10 days for the rest of your trip.
The best thing to do is start reseraching where you want to go and plan your trip in detail. Find out WHAT you want to see (TripAdvisor is great, just check "Things to do in-------".)
I can tell you that driving a rental car in and around London is a painful experience. Southern England has heavy traffic as well. Just driving 75 miles in a day on scenic roads may take you over two hours. Then parking can be difficult in some place, especially large cities. Look for lodgings where parking is provided.
Next, remember if you just set out to do for example, England and Scotland, then that would clearly include London and Edinburgh. That takes up HALF of your trip. Nothing wrong with that, but the countryside is amazing in both countries.
The Cotswolds are great, we stayed at Chipping Campden that don't have a rail connection to anywhere. We had a rental car. We used Chipping Camden as a base to do the Cotswolds (2 full days), Oxford, Blenheim Palace and Stratford Upon Avon. We were there six nights and had five full days for touring.
One of our favorite places to visit in Great Britain was York. It is very historic with an amazing Minster (Cathedral), ancient walls that date back to the Romans and much more. We did three nights there and were busy the entire time.
There are many, many great places to visit, just figure out what you want to see, plan it out and don't try to overdo it.
Another issue, you want to keep it budgeted at $6,000 for four of you. That will be extremely difficult. Places like Premier Inns in cities like London are reasonable, but with four persons you many need two rooms. We love B&Bs and find them cheaper than hotels, and you usually get a free breakfast. Also, if you rent a car, you will need a car large enough to hold luggage for all four persons.