As a counter argument, even if your possession of an accent (and a thick one, by your own admission) was considered a protected element of nationality, the employer could still deny you on those grounds if they meet three criteria:
The requirement must be occupational in nature, with a link between the job and the requirement
The requirement must be in service to an articulated legitimate aim.
The requirement must be proportionate to the articulated aim, so that the restriction must be a reasonable way of achieving it.
So, it depends on the position applied for: if its is, for example, a janitorial job, its probably not legitimate, but if its a bank teller, phone center caller, or other verbal communication position, especially a "customer-facing" one, this may be a legitimate requiriement.