Has Lane ever been to the UK? It's not a description that I recognise, despite having lived here all my life!
Late July and all August is when the schools are off and therefore places are busiest. Few people with families have camper vans! These are far more likely to be owned by retired people, who don't holiday when the schools are on holiday.
Oddly, the weather is often better here in May than in June, although this year and last, both have been wet. September weather is also good, which is when I would travel or the first 2 weeks of June. Daylight hours are slightly longer in June, but still good in September.
As has already been stated, there are 2 Bank Holidays in May - the 1st and the 29th, so places will be busier then and more expensive. The Lake District and Cotswolds will have crowds from Easter until early October so are best avoided if you don't want crowds - there are many other attractive places to choose from here.
Shops in the UK don't close for holidays, unless you find a shop owned by just one individual with no staff to stand in whilst they are away, but this will not impact on your holiday. This isn't France!
How far out to book accommodation depends on where you are going - e.g. there will be lots of choice in London, but you will need to book Skye months out. I usually book a September holiday by early January. Booking.com and Airbnb are sensible websites to look at. If you want good, basic rooms in cities or on major roads, consider Premier Inn or Travelodge chains. Sawdays.co.uk and smoothhound.co.uk both have a good selection of B&Bs and inns. There are other websites if you want a 7 night cottage rental.
I have never found much to choose between car hire companies. I use the consolidator Holiday Autos. Note that some more remote car hire places aren't open on a sunday. The default hire is manual transmission. You can specify an automatic, which will be more expensive, but I don't think they guarantee this. Don't drive straight off a transatlantic flight.