This is a tricky question and from my own experience, varies from school to school. Here's just one example; I was offered a position at a school once, which offered the DELTA to its employees part-time over two years. I asked if it would be possible for me to do it, and they said that it would be very unlikely as I didn't have a degree. In fact, it was as if they had already decided to fail me. The attitude was so negative.
I questioned this need for a degree, and they said that graduates were more used to the workload. Well, I knew one or two people who had previously done the DELTA who did not have a degree.
So, there you go. Even if the Cambridge ESOL website doesn't specify the need for a degree in order to do the DELTA, many places will insist that you do.
My advice... perhaps just start a degree and play it by ear. You could even find a school that offers the DELTA to its employees and if they let you, do the DELTA while studying for a degree. That way, you could have the best of both worlds.
Good luck :-)