I've been on MSc degree (MS1), however due to health issues, I haven't finished dissertation and got a long extension. I have applied for another MSc (MS2) at the same university and which suits my interests better. The problem is that the admission team told me I can't be enroled both for MS1 and MS2 (is there a way to go around it? It's a Scottish university).

The options seem to be: either I quit MS1 early and get a Diploma, thus can enroll on MS2 or I work in the upcoming months to finish MS1 and re-enroll to do MS2 next year.

The thing is, MS1 with a dissertation part won't conribute much in terms of employability, though it would help a bit in terms of getting onto PhD. However, PhD I'm interested in does align much better with MS2 and it would be a better predictor for successfuly applying for PhD to have MS2 than MS1, however good the dissertation relating to MS1. I'd rather have spent one more year only before applying for PhD as I'm already in my thirties, then spend two more years to finish MS1 and then MS2.

Any suggestions what course of action should I take or anybody had a similar dilemma in the past?

Seems like your best option is to take the diploma for MS1 and do the MS2. There's not much point in doing both. Or you could just finish the MS1 and apply for the PhD.