To start with, you will want to make sure that your passports are not going to expire within six months of the end of your anticipated period of travel.

Although for normal tourism, as a U.K. resident, you would enter under the visa waiver program, due to the extended nature of your proposed trip you are going to need to apply for a B2 tourist visa. A B2 visa can be authorized for up to six months, and can be extended for an additional six month period to cover the full duration of your travel. However, you cannot circumvent the need for an extension by going to Canada just shy of the expiration of a six month stay then trying to reenter the U.S. on a new B2 visa - if you don't have an extension and don't actually leave North America before attempting to reenter, you can anticipate that you'll be denied a new I-94.

Conventional wisdom is to apply for an extension of your visa not less than 90 days after entering the U.S., to avoid sending the signal that you intended to stay for more than the maximum six month period associated with the original application, but not less than 45 days before your visa expires due to the amount of time it takes to process the request for extension. That seems a bit odd under your circumstances, given that you are explicitly planning a tour greater than six months in duration, but... you have bureaucracies in the U.K. as well, so you know about government agencies and their rules. The rationale for the six month visit for pleasure tourism is that very few people can afford to go more than six months without working; it's easier to get an extension if you're entering for medical treatment that's anticipated to take longer than six months or as a business tourist (as your non-U.S. employment is a continuing source of income).

The application for an extension is requested with Form I-539, and you would also want to read Form M-752 (it's a very quick read), both available here.

As an aside, you may want to explore the possibility of leasing a RV in the U.S. - it's odd enough to drive "on the wrong side of the road" when the driver's seat is on the left side of the car. I'm not sure how the cost would compare to shipping a vehicle to and from the U.K., and operating it for a year within the U.S., but there's also a safety angle. Also, if you prearrange a return shipping date you should consider how that might look to an immigration officer if you are initially approved for only a six month visit.