Israel, the problem is there are two answers (setting modes aside as the question is about tonic solfege). The complex one is the correct one: firstly you translate the key signature to solfege (C=Do, D=Re etc). Then you translate the note within that key to a relative from the tonic (the key signature), so a second is Re, a third Mi, etc. If you change the key signature, then you get an instant transposition as a result, in theory - of course in practice, because the sound of each key can be very different, you may well end up with a different emotional weight, however. But this is more a theoreticians game than what you'll come across out there.
The simpler solution is to presume a key of Cmaj, so all other keys are modal, and the notes are simply renamed, C:Do, D:Re, E:Mi, F:Fa, G:So, A:La, B:Ti, with the relevant sharps/flats. This is what you'll most commonly use, and - I think - what you're looking for, incomplete though it is.