It is not taxed (generally), as it is part of the 'special scheme' and so disregarded for tax purposes.
The Local Authorities (Members' Allowances)(England) Regulations 2003, specifically travel, travel expenses and the catch-all "any other expenses which are wholly, exclusively and necessarily in the performance of their official duties".
If the allowance, is not set out as being to allow the office holder to perform their duties, i.e. it is more a salary than it is to cover the expenses of office, then it is taxable.