The size of proofs is almost totally unrelated to the size of the statements they prove (in the worst case).
Lower Bound: constant size
The size of the smallest proof for statements of size N is constant. Just make your statements of the form "True or S". (Unless you count the input statement as part of the size, in which case it is linear.)
Upper Bound: incomputable size
The size of the largest proof proving a statement of size less than N grows so fast it is not computable from N.
Consider the size of a proof that a Turing Machine does not halt within S steps. Some turing machines run forever but can't be proven to run forever. Suppose you have such a machine M. Proving M doesn't halt before S steps must take more steps as S grows. Otherwise we have a proof for all S and have contradicted ourselves.
Lets call that number-of-steps to proof-size function G. I am going to make one assumption, which is that there is a computable function G' such that G(G'(X)) >= X.
Suppose you have some computable function F you claim is an asymptotic upper bound on proof sizes given the statement size. Then I just generate a sequence of statements where the Nth statement is "M does not halt within G'(N*F(N)) steps". The statements are all provable (worst case: simulate M in the proof; that's why I assumed G' is computable) and grow in size logarithmically. But the proof sizes grow asymptotically faster than F, which is exponentially faster than F with respect to the statement sizes. So F is not actually an asymptotic upper bound.
Therefore no computable function is an asymptotic upper bound on proof size given the statement size, which means the upper bound is incomputable.