If you want to consider smooth (weak) Fano variety, then Fukuda has effective estimation of the birationality of anti-canonical systems for any dimension (but not optimal), see [S. FUKUDA, A note on Ando’s paper “pluricanonical systems of algebraic varieties of general type of dimension ≤ 5, Tokyo J. Math., 14(1991) 479-487.]

If you want to consider Fano variety with at worst canonical singularities, then we know the answer up to dimension 3, see my joint work with Meng Chen, [On the anti-canonical geometry of Q-fano threefolds, J. Differential Geom. 104 (2016), no.1, 59--109]

If you want to consider Fano varieties with klt or lc singularities, there is so far no explicit answer to your question even in dimension 2.

Of course I should mention that the existence of such effective birationality for $\epsilon$-lc Fano is proved by Birkar last year, but his method does not give a way to estimate the number $m$.

By the way, if you are interested in Fano varieties with klt singularities which is K-semistable (e.g. admitting Kahler--Einstein metrics), then in this case we have effective estimation of $m$ by my recent work, see proof of Theorem 1.5 of arXiv:1705.02740.