Zhen Lin Low's paper _Universes for category theory_ (arXiv:[1304.5227](https://arxiv.org/abs/1304.5227)) is perhaps a great start. The abstract is:
>The Grothendieck–Verdier universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-sets or even the category of all locally U-small categories, where U is an “arbitrary but fixed” universe, all without worrying about which set-theoretic operations one may legitimately apply to these entities. Unfortunately, as soon as one allows the possibility of changing U, one also has to face the fact that universal constructions such as limits or adjoints or Kan extensions could, in principle, depend on the parameter U. We will prove this is not the case for adjoints of accessible functors between locally presentable categories (and hence, limits and Kan extensions), making explicit the idea that "bounded" constructions should not depend on the choice of U.
What this means is that for large categories that are built in a sensible way from small categories, certain universal constructions (like limits) in a 'truncated' version of the large category relative to a universe keep their universal property after being included to a larger truncation.
There are also [slides of a talk][1] by Low on the same topic that are a bit more ... accessible. I want to point out the concept of "weak universe" suggested at the end of the talk is more or less the same thing as some stage $V_\alpha$ in the cumulative hierarchy where $\alpha$ is some limit ordinal (and also not too different to a model of [ETCS](https://ncatlab.org/nlab/show/ETCS) - although I'm deliberately being vague here). Such a $V_\alpha$ is a more general (weaker) notion than the $\mathbf{U}$ in your question. No results are given regarding weak universes, but keep them in mind for the rest of this answer.
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From the point of view of algebraic geometry, the [Stacks Project](http://stacks.math.columbia.edu/) goes to great lengths to make sure [everything is a set](http://stacks.math.columbia.edu/tag/0009). In particular the proof of the [Reflection principle](http://stacks.math.columbia.edu/tag/000F) is unwound (or reverse engineered) to show how instances of it apply in all the cases of interest for categorical constructions involving schemes. Note that the Reflection principle applies involves the sets $V_\alpha$ for limit $\alpha$, hence weak universes. The trick is to write down an explicit (super)exponential function $Bound$ in cardinals (like $\kappa^\kappa$ or $\kappa^{2^\kappa}$) that bounds the growth of the cardinality of the sets necessary to build the various scheme-theoretic constructions on wants, and then use this to show the (small!) category $Sch_\alpha := Sch_{V_\alpha}$ is closed under said constructions (see eg [Tag 000R](http://stacks.math.columbia.edu/tag/000R)). So for example, one can get existence of pullbacks and $\leq\lambda$-sized colimits (for a regular cardinal $\lambda$ - the notes prove the case of $\lambda = \aleph_0$; a simple modification of the bounding function allows for larger but fixed $\lambda$). [Also treated](http://stacks.math.columbia.edu/tag/000W) are _sites_, which of course need to be small to make sheaf categories locally small.
All of the above would be in vain, but for the fact that all the constructions one wants to show are in your small category of schemes are in fact performed in the large category $Sch$ of _all_ schemes, and then the theorem shows that these are actually in the (essential) image of $Sch_\alpha \hookrightarrow Sch$, for some appropriate $\alpha$. Given any concrete problem at hand (for instance, discussing the geometry of the compactified moduli stack of $n$-pointed genus-$g$ curves) one only ever starts with a set's worth of data, and so there exists some limit ordinal $\alpha$ such that no consideration of anything other than $Sch_\alpha$ is necessary. In particular, not even full universes are needed in this instance, although using universes might make the book-keeping of the proofs simpler.
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If you are more interested in differential geometry, then the category of all finite-dimensional manifolds (of your preferred flavour) is already essentially small. This is due to the Whitney embedding theorem.
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From a topological point of view, many interesting categories of spaces (even if not excessively general) are already essentially small, while being closed under a reasonable class of (co)limits. For instance, second-countable paracompact Hausdorff spaces all embed in the [Hilbert cube](https://en.wikipedia.org/wiki/Hilbert_cube#Properties) $[0,1]^{\aleph_0}$, hence the category of such is essentially small. Polish spaces are another example of spaces embeddable in the Hilbert cube. One could perform a similar analysis to the Stacks Project for topological spaces in any given setting to find out which constructions allow one to stay entirely within a small category.
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Lastly, it should be worth pointing out that in the category of sets, one actually has reasonably canonical constructions of lots of things, for instance a limit [can be given as a specific set of natural transformations](https://ncatlab.org/nlab/show/limit#in_), rather than just positing that such a set exists with some universal property. Likewise for colimits; given powersets (which in ZF(C) are unique, for instance) there is a construction of a colimit using Separation as a subset of an appropriate powerset. Then the universal property follows from the construction, so the comparison to arbitrary-sized sets is no longer a problem (in the sense that, going up a universe cannot destroy the universal property).
Then for various categories of algebras for monads on $Set$, unless things are really weird (laptop is running low on battery, I've not time to remind myself if it's _always_ ok: you might need the monad to have a [rank](https://mathoverflow.net/a/55356/4177)) these type of results lift, if I'm not mistaken.
[1]: http://zll22.user.srcf.net/dpmms/slides/2013-07-11-AccessibleAdjoints.pdf