Having worked at 9 different units in the UK I have seldom found accent a problem on the R/T. Often the Scottish and N Irish ATCOs come across better on the R/T because of the way they project their voices. South London is bad for co-ordination in a noisy room because it is usually delivered in a mumbling tone.
What is a problem is local dialect and pronunciation.
The only trainee I have ever thought would be hindered by his voice had a very pronounced local way of emphasising different parts of a word which often ended in a tailing off at the end. In six weeks working with me he had so many calls of "say again ???" that even he knew there was a problem.