Independent travel is not allowed in Bhutan. British passport holders are required a visa when entering Bhutan.
You can directly get in touch with local tour operators in Bhutan via their Tourism Council of Bhutan website, or you may have your travel agent arrange for everything on your behalf. Your local tour operator will be the one to submit your application for a visa. Visa clearance must be secured from Thimphu prior to arrival in Bhutan. Once you have been cleared, that is the only time that you will be able to purchase plane tickets going to Bhutan. This visa clearing takes about 10 days to complete. You will be charged 20 USD for this visa, and any extension to your 15-day stay in Bhutan may be arranged for an additional fee.
The only airline servicing Bhutan is Druk Air, and flies in the Paro International Airport from India, Bangladesh, Nepal, and Thailand only. Tourist packages have government set tariffs which are non-negotiable, and you would need to pay for these costs prior to arriving in Bhutan.
Should you wish to contact local tour operators to arrange your trip to Bhutan, you may access more information at their website.