I'd think that mathematically the Galilei-Newton spacetime manifold is a bundle. You have time along which you have a 3D Euclidean affine space pinned at any point of this (oriented) time axis.

Of course you can also get the Galileo-Newton spacetime by "deformation" of Minkowski spacetime, which is formally given by expansions around ##c \rightarrow \infty## (##c## speed of light in vacuo). This is sometimes useful when taking the non-relativistic limit of relativistic equations.

This preprint "On the Ultrarelativistic Limit of General Relativity" by G. Dautcourt https://arxiv.org/abs/gr-qc/9801093
also looks useful with its section on differential geometry with a degenerate metric.

My interest in this line of thinking was motivated by this paper:
"If Maxwell had worked between Ampère and Faraday: An historical fable with a pedagogical moral"
by Max Jammer and John Stachel
American Journal of Physics 48, 5 (1980); https://doi.org/10.1119/1.12239

Formulate (possibly secretly and not necessarily with all the details laid out to the end-user)
the physics of PHY 101 in a Galilean-spacetime way (or at least a spacetime-friendly way)
so that the transition to Special Relativity is smoother and less drastic (rather than the more common way...
suggesting that much of what was learned must now be unlearned (or quickly refined) to understand Special Relativity).
While the history of how we came to understand relativity is interesting,
as JL Synge puts it...

...alternatively, one can try a similar pre-"Quantum Physics" formulation.