The Trust ... was founded as a not-for-profit company in 1894 but was later re-incorporated by a private Act of Parliament, the National Trust Act 1907. Subsequent Acts of Parliament between 1919 and 1978 amended and extended the Trust's powers and remit. In 2005 the governance of the Trust was substantially changed under a Scheme made by the Charity Commission.
from wikipedia
It is definitely the case that the NT is & has been granted a number of privileges by the government. For instance the Wikipedia article mentions the right to declare land "inalienable", which prevents the government passing a compulsory purchase order on the grounds of a historic house in order to build a motorway (it may grant them wider-reaching powers than this, I'm not sure).