This is a riff on this query, concentrating on aspects of functional analysis and related topics. It is formulated in a conversational manner but is based on the concepts of ind and pro categories which were introduced in the 50´s by Grothendieck and collaborators as a method to add limits and colimits to a given category and can be made precise in this context. I have added the intermediates cases of limits for countable spectra in the pirit of this question.
In fact, I think that it is sometimes advantageous to consider a threefold transition--from the finite to the countable, then to the uncountable.
Thus we can start with the category of finite dimensional Banach space (with linear contractions as morphisms)- This leads successively to those of separable, resp. non-separable spaces (ind-categories).
Dually, one gets the separable Waelbroeck spaces, resp. Waelbroeck spaces (H.Buchwalter and L. Waelbroeck).
Starting with the Banach spaces (now with continuous linear mappings as morphisms), one proceeds to Frechet spaces, respectively locally convex spaces as projective limits. Dually we obtain, with the pro-construction, convex bornological spaces of countable type (closely related to Grothendieck´s $DF$-spaces) and convex bornological spaces.
If one applies the pro construction to the category of Banach spaces, with contractions as morphisms, one gets the class of Saks spaces, initially those with metrisable unit balls.
Analogue constructions can be carried out in the categories of Banach algebras (resp. $C^\ast$-algebras), commutative or non-commutative, with or without units.
In topology, one can start with the compact subsets of euclidean pace as the finite objects, then proceed to the compact metrisable spaces, followed by the compact spaces.
Using the ind constructionand starting with the category of compact spaces, one obtains the compactologies, initially those of countable type (again Buchwalter and Waelbroeck). These are a natural parallel to the category of topological spaces but have more convenient properties. One can move back and forward between these two categories so that they coincide for a large fraction of the topological spaces which occur in practice--metrisable space, locally convex spaces, more genereally $k$-spaces wich means that their theory coincides in many contexts.
Touching on an example mentioned in the query, one can start with metric spaces with Lipschitz functions as morphisms--the pro-construction turns up the category of metrisable uniform spaces, followed by all such spaces.
This is a small selection of the kind of spaces which arise in this manner. It could be extended almost to infinity. I have concentrated on examples which have already been studied intensively in their own right but there are many where this is not the case, but perhaps should be.
Of course, the thrust of the question is on the extension of results and proofs from the countable to the uncountable case. There are so many such examples in the above situations that I can only realistically with one and I have chosen that of tensor products. Suppose that we have some notion of a tensor product for finite Banach spaces (I am thinking of the inductive and projective norms but the following holds much more generally). Then one can extend it successively to the categories of Banach spaces, of locally conovex spaces and of convex bornological spaces in the natural manner.
Another recurrent situation is to take a standard duality from a special situation (typical examples--Riesz representation or Gelfand-Neumark duality for the case of compact spaces) and extend it to one for general topological spaces (more precisely compactologies but this can be used to deduce results for the topological case using the above mechanism)