I am planning to apply for the same MSc program at different schools. As of now I have only completed a single application, and that school has accepted my application. Would it be good for me to mention in my subsequent applications that I have been accepted to another school for the same program?
I doubt that it would do you any good in most cases and it might do you harm. There may be some upside, but consider the down. People might decide that you aren't very interested in them if you mention another school. People might decide you are gaming them. But even the best case is that people might decide that you already have options so they would want to give similar options to others. A "better" school might wonder why you applied to a "worse" one. A "worse" school might conclude you won't accept an offer from them.
It just seems too risky to me.
Focus your application on what you have done, on your interests, and on those things that make you a candidate for success in the program you are applying to. If you are a strong candidate it will be assumed that others will evaluate you similarly. You aren't desperate, so don't sound as if you are.
On the other hand, I don't think you need to actively hide your other acceptances. If asked, explain why you would prefer the new institution.