"Famously, one of the objections against Newton's theory of Gravity was that it instantaneously acted at a distance."
Firstly, historically, I'd say it's not necessarily an objection against Newton's gravity but against special relativity. Secondly, it's worth pointing out that the problems in considering the two theories next to each other are not merely of philosophical nature. If two observers in special relativity move with non-vanishing velocity relative to each other, their conception of spatial now is different, geometrically tilted to one another. And if you have instantaneous action, you might be able to affect events in the others past. The concept of the Tachyonic antitelephone is similar.
The solution was found in Einstein's GR, where it was seen that it was spacetime itself that propagated the influence.
I have some minor issues with some formulation. E.g., the expression, "propagated the influence", seems to take one concept and make two out of it for the sake of rhetoric.
Also, and this is just an digression, I always like to point out that there were earlier attempts to do geometric gravity. And many later attempts, although they were never able to overtake Einstein's construction. The alternative proposals seem endless. This list contains some classical ones.
"Now one of the insights of the mathematical investigation of space, is that there are two structures - the space of points and the space of contiguities ie its topology. This means that distant points can be connected directly without there being a propagating influence."
Again, minor issue: To a contemporary mathematican, calling a mere set of points a structure sounds like calling a group of people (e.g. Madonna, Obama, Bob and you) a team. Dedekind said: "I imagine a set to be like a bag of things."* And then such a set is considered a space as soon as you equip it with a topology.
This is still a broad notion compared to what one would usually take to be space. When you speak about propagation of the gravitational effect, the sense in which "points can be connected" by topology is rather loose. Topology is able to capture a sense of neighborhood. This MathOverflow thread is of interest, but it is also a lot broader than that.
Connectedness is a feature. You say they can be connected directly, but without some additional structure, such as a notion of distance, this sort of connection has not much effect. Conversely, a distance function (metric) induces a topology, but the topology induced by a classical metric hardly leaves space open for playing around.
*Cantor said: "I imagine a set to be like an abyss".
Does this mean that one should consider a radically different topology on quantum reality (which in Bohmian Mechanics is real) such that every point is connected? But in conventional topology this means that the topology is trivial.
By definition, the whole set is part of the topology in any case.
Does this then mean one needs a 'quantum topology' that incorporates somehow the mechanics into quantum reality so that the topology is not trivial?
Again, the notion lets one restrict the class of functions (one can restrict oneself to investigate continuous functions, a concept which depends on the chosen topology) but a topology alone doesn't do much on its own.
Contemporary theories are build upon spaces which admit a map to R^n, and a spacetime metric like general relativity produces topologies which are very classical. Nobody really knows how to improve this successfully, but of course, there are people who are of the opinion that the problem of matching the quantum world with the classical space concept is by not starting with space in the first place. I can't really say anything about the question if there is a natural topology for, e.g., emergent gravity scenarios which differ much from the ones which are well investigated. I also don't know to what extent the notion is even necessary if, e.g., spacetime is just the spectrum of an abstract operator. I think noncommutative geometry goes along these lines.
Lastly, although this kind of work is arguably more far fetched than the already highly hypotetical attempts above, the authors of this paper argue for non-set foundations for physics and in the first two chapters, they give their arguments.