My own BSc (Hons) is certainly not needed for the job, but good expereince at Uni, good prep for the NATS Student ATCO Course, (though at times it made the BSc course feel like a 4-year holiday), and always good to have the qualification as a back-up/Plan B to use in the event that the Student ATCO course doesn't go the right way. (Pretty high rate of attrition in my day).
Experience is all well and good, but your ability to successfully complete the aptitude tests coupled with how you conduct yourself in an interview will have a greater impact. You'll then have the Student ATCO course to get your Licence and the subsequent posting to your unit to Validate it to look forward to - which will likely take 2-3 years, and a good number of those bums on seats on Day 1 will not make it to validating their Licence.
Me? I'd do the course, get the BSc (Hons) under your belt, then apply for ATCO. Others may tell you different.