Without "uniformly on $\alpha$" and assuming $M$ is a smooth manifold (i.e., the maps $\phi_\alpha\circ\phi^{-1}_\beta$, where defined, are smooth), this appears to be the definition of a $C^k$ embedding of $M$ into $\mathbb{R}^N$.

I don't know of any standard terminology if "uniformly on $\alpha$" is assumed, but one could call such an embedding a "uniformly $C^k$ embedding".

Any compact manifold $M$ certainly admits such an embedding, and, even if you put no constraint on $N$, I'm pretty sure any manifold $M$ has a uniformly $C^k$ embedding.