  We first establish new local limit estimates for the probability that a
nondecreasing integer-valued random walk lies at time $n$ at an arbitrary
value, encompassing in particular large deviation regimes. This enables us to
derive scaling limits of such random walks conditioned by their terminal value
at time $n$ in various regimes. We believe both to be of independent interest.
We then apply these results to obtain invariance principles for the Lukasiewicz
path of Bienaym\'e-Galton-Watson trees conditioned on having a fixed number of
leaves and of vertices at the same time, which constitutes a first step towards
understanding their large scale geometry. We finally deduce from this scaling
limit theorems for random bipartite planar maps under a new conditioning by
fixing their number of vertices, edges, and faces at the same time. In the
particular case of the uniform distribution, our results confirm a prediction
of Fusy & Guitter on the growth of the typical distances and show furthermore
that in all regimes, the scaling limit is the celebrated Brownian map.
