You subscribe or take out a policy.

annulation : one "l"
cancellation : two "l"s

If the company concerned is an insurance mutual, then there is always an idea of membership behind it. This probably explains the use of the word "adhésion".

http://www.namic.org/

US Site buy the explanaiton equally valid for UK contexts.

The mutual insurance concept originated in England in 1696 with the establishment of the first mutual fire insurer. The idea migrated to America with the creation of the Union Fire Company in Philadelphia, Pa., in 1735, founded by Benjamin Franklin, no less.
Unlike stock insurance companies, which are owned by investors who may have no other connection with the company, mutual insurance companies are owned by their policyholders. Policyholders are entitled to vote for members of the company’s board of directors, and may receive special dividends in the form of capital distributions or reductions of policy premiums. Unlike stock companies, mutual companies exist solely to serve the insurance needs of their policyholders, and not to provide investment profits to shareholders.
Mutual insurance companies range in size from small companies operated out of managers’ homes - providing insurance to America’s farmers to protect the dwellings, buildings and machinery that make farming possible - to some of the largest insurers of homes, automobiles and businesses in the world.