If you can speak English well enough to be understood, and meet the required ICAO level of proficiency, that is good enough. I've never heard of anyone losing out on a job because of their accent.
And Banana Joe:
Some of us Nigels are stuck with proper Esssix accents, and proud of it. That said, having just read the requirements for ICAO level 6, I'm really not sure I meet the standard.