2nd master's degree in "slightly-different" field? data science and machine learning specifically So i was accepted to a master data science program in the UK, but I am also interested in a machine learning specific program at another institution, also in the UK. 

The thing is that my data science program has some overlap to the machine learning program, but not completely. The machine learning program has significantly more in-depth courses, and many other courses regarding deep learning's applications that are not available at my current program.

So is it fine if I apply for the machine learning program after the data science one? In the UK, are there rules that forbids this? Or would the admission committee just throw my app out since they deemed for some reason the two programs are too similar?

Thank you.
I don't know of any generic rules which forbid this. Indeed taking a masters degree after another higher degree is an accepted, although uncommon, method to switch career paths fairly deep into one's education. On the other hand, as you note yourself, that's not really what you are suggesting here, and any application is going to have significant barriers to overcome, firstly as to how you'll be funding this (masters courses are expensive, after all) and secondly as to why you aren't following the more natural career path of moving on to a Ph.D. if you want to continue your studies. Which isn't to say that a sufficiently strong narrative wouldn't manage to get you an offer if numbers weren't limited and you could explain your funding.

(It's worth noting that while the University probably doesn't have a rule against this, many potential sources of funding do)