For $C^{1,\alpha}$ convergence, it is enough to have a uniform upper bound on diameter, a uniform lower bound on volume, and a uniform bound on sectional curvature. This is due to Cheeger and Gromov. The proof works for Riemannian manifolds for any dimension. For the case of collapse, I think your question is subsumed by the papers "Collapsing Riemannian Manifolds while keeping their Curvature Bounded" parts I and II, by Cheeger and Gromov, but you are probably more interested in simplifications of this general result in your case.

You may be interested in corollary 2.3.1 of Heintze-Karcher's paper `A General comparison theorem with applications to volume estimates for submanifolds.' They give an explicit version of Cheeger's inequality, in that you can put an explicit lower bound on the diameter if you know the length of the shortest geodesic, the volume of the surface, and a bound on the curvature.

I know you have said that you are familiar with Anderson's work, but in section 5 of his paper "Cheeger-Gromov theory and applications to general relativity" (math.sunysb.edu/~anderson/cargese.pdf) he seems to indicate that this is basically an open problem, and describes why using a curvature bound without choosing coordinates can be unsatisfactory. Namely that he describes the non-compact class of "plane-fronted gravitational waves" that satisfy $|R|^2 = 0$. Of course it would also be nice to hear about newer, updated work if it exists.

Dan- thanks for the clarification. The definition of smooth structure in use here is what I would call a 'diffeomorphism type'.

@Mariano I don't see how one gets a local diffeo here. For instance if I choose $(\mathbb{R}$, $f(x) = x^3$, d$x^2)$ as my manifold, smooth structure, and metric, then the unit speed geodesic emanating from $0$ is just the identity map. This does not give a local diffeo at $0$ from $(\mathbb{R}, \textrm{Id})$ to $(\mathbb{R}, f)$ since the transition map is not differentiable at the origin.

Perhaps it is worth clarifying your definition of smooth structure? For instance the smooth structure determined by $(\mathbb{R}, f)$ where $f(x) = x^3$ is usually given as an example showing that $\mathbb{R}$ does not have only one smooth structure.

@Willie the book is possibly the first in a series, it is called "Physics for Mathematicians, Mechanics I" (amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/‌​dp/…). A free early version is also on Shifrin's website (math.uga.edu/~shifrin/Spivak_physics.pdf) where it is called "Elementary Mechanics from a Mathematician's Viewpoint"

Spivak's "Calculus on Manifolds" is reasonably approachable for an undergraduate, and Ted Shifrin has some excellent lecture notes on elementary differential geometry on his website (math.uga.edu/~shifrin/ShifrinDiffGeo.pdf) that he uses to teach an undergraduate course on differential geometry of curves and surfaces.

Your comment reminded me that all of this is explained beautifully in an address Minkowski made in 1908 called "Space and Time." I haven't found it online but it is published in the Dover book titled "The Principle of Relativity." In the article Minkowski gives a mathematical way to go from Newtonian mechanics to SR and also argues that one should not treat space and time separately.