PhD Application including President 's scholarship at Imperial College

I applied for a PhD in Mathematics admission at Imperial College October, 2022. I have not heard from the department, Do I need to contact the department? Or the admission was rejected?

Were you in contact with a potential supervisor, that is really a key thing for a scholarship application and your best source of information on how an application is progressing.

I was unable to get a supervisor as at the time I applied, so I just submitted the application

Sorry, but that almost certainly means that you were not put forward by a research group for a scholarship, that scheme is hyper competitive and goes through several rounds of review at group and then department level before a few get sent on to College. Yes, you should contact maths about your application but if you are not a UK national funding is very unlikely at this point I am afraid.