First let's see if we can make F, G and U independent of each other by using Solve
Solve[{F=f*c*G, G=D+F/(1-f)+U, U=(1-f)*G},{F,G,U}]
That neatly separates the F, G and U and gives us, among a couple of other things,
F=c*D*(f-1)/(c+f-1)
G=D*(f-1)/(f*(c+f-1))
U=-D*(f-1)^2/(f*(c+f-1))
We are very fortunate that all those are expressed in terms of the variables you are looking for.
Next eliminate the T in your original equation N=D-T+F using your given T=P-F
ReplaceAll[N=D-T+F, T->P-F]
which gives us
N=D+2*F-P
and then eliminate the P in that using your given P=G*t
ReplaceAll[N=D+2*F-P, P->G*t]
which gives us
N=D+2*F-G*t
Now eliminate the F in that using the result from Solve that we did above
ReplaceAll[N=D+2*F-G*t, F->c*D*(f-1)/(c+f-1)]
which gives us
N=2*c*D*(f-1)/(c+f-1)+D-G*t
and then eliminate the G in that using the result from Solve that we did above
ReplaceAll[N=2*c*D*(f-1)/(c+f-1)+D-G*t, G->D*(f-1)/(f*(c+f-1))]
which finally gives us
N=-D*(f-1)*t/(f*(c+f-1))+2*c*D*(f-1)/(c+f-1)+D
It seems like there should be a simpler and less error prone way of doing this where Wolfram|Alpha does more of the work in a single step. Perhaps someone else can see a way to do that or perhaps in a day or two you or I might realize a way to do that.
Please check all this VERY carefully to make certain I haven't made any mistakes.