I travel to the UK and Europe about once a year due to my partners family being there. We usually rent a car. We always book a car though Auto Europe in Australia, seem to be good prices and puts a middleman between us and the sometimes questionable business practices of some car hire companies. Be aware that uk car hire companies charge extra for a second driver (this can be about $20 a day!) while it is sometimes included in hires through Auto Europe - something I look for. I have no connection to Auto Europe other than being a satisfied customer. If staying in London for a few days - you do NOT want to be saddled with a hire car - arrange to pick it up when leaving.
I generally find campervans to be quite expensive and by the time you add in extra fuel costs and campsites there are not big savings to be had.
We usually book accommodation in London but wing it the rest of the time but some of the hotel chains have very good prices if booked well in advance. Fortunately we own a small flat near my partner's mother where we base and do forays out for a few days at a time. Packing up each morning gets a bit "old" quite quickly. If you can find a cheap place for 2 to 4 days and day tour from there, then move on to another base location.
Get a UK sim for the phone when you get there - you will need it to sort out accommodation etc. There will be a couple of shops selling them at Heathrow - often offering plans from multiple networks (3, EE, Virgin etc). I find a monthly pay as you go plan with about 3 gb of data allows the phone to used for navigation in the car. Cost about $25-30 (15 pounds).
The cheapest time of year is when it is cold and raining

Oops - thats most of the time! Best to aim at the shoulder seasons around May or September/October. Mid winter is pretty dire and the days very short.
"Perfection is attained not when there is nothing more to add, but when there is nothing more to remove".