I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things that can be said at a first glance:
Manifolds would not necessarily admit partitions of unity, so not every Manifold would admit a Riemannian metric.
A Haussdorf manifold admitting a connection would be automatically second-countable.
A Haussdorf manifold admitting a Lorentzian metric would be automatically paracompact.
Manifolds without these two assumptions wouldn't necessarily have the homotopy type of a CW-complex.
Since the question is maybe too broad, we can focus for example on the following especific issue:
How Berger's classification of manifolds of special holonomy would change without these two assumptions?
One motivation to ask this question is that manifolds describing space-times needn't to be Haussdorf, or at least physicists have studied non-Haussdorf solutions to General Relativity, see for example "The large scale structure of spacetime" of Hawking and Ellis.