Fundamentally, working in NBG is not much different from working in ZFC, except that you are allowed one level of freedom to form collections of sets that are not themselves sets.
As such, you still have to encode the actual "elements" you want to work with as sets, and the encoding has to be reasonable enough that all the "operations" you wish to perform on "elements" can also be encoded.
Conversely, if you can do such an encoding, then there is probably no obstacle to working in NBG.
Regarding the existence of localisations: there is basically no obstacle in NBG.
Let $\mathcal{C}$ be a category in NBG, by which I mean we are given classes $\mathcal{C}_0, \mathcal{C}_1, \mathcal{C}_2$ and maps $d_0, d_1, d_2 : \mathcal{C}_2 \to \mathcal{C}_1$, $d_0, d_1 : \mathcal{C}_1 \to \mathcal{C}_0$, $s_0 : \mathcal{C}_0 \to \mathcal{C}_1$ satisfying the simplicial identities and the strict Segal condition.
Let $\mathcal{W}$ be some subclass of $\mathcal{C}_1$; for simplicity, assume it contains all identities in $\mathcal{C}$ and is closed under composition.
We can construct $\mathcal{D} = \mathcal{C} [\mathcal{W}^{-1}]$ as follows:
$\mathcal{D}_0 = \mathcal{C}_0$.
$\mathcal{D}_\rightsquigarrow$ is the class of finite sequences of elements of $\mathcal{C}_1$, say $(w_0, f_1, \ldots, f_n, w_n)$, where each $w_i$ is in $\mathcal{W}$, $d_1 (w_i) = d_1 (f_{i+1})$, and $d_0 (f_i) = d_0 (w_{i+1})$; in words, this is the class of zigzags where the backward-pointing arrows are in $\mathcal{W}$.
We define a binary relation on $\mathcal{D}_\rightsquigarrow$ relating two zigzags if there exists a small (i.e. set sized; actually, we could even assume finitely generated) subcategory $\mathcal{C}' \subseteq \mathcal{C}$ containing all the morphisms in the zigzags, such that the zigzags become equal in $\mathcal{C}' [(\mathcal{C}' \cap \mathcal{W})^{-1}]$.
It is easy to see the above is reflexive and symmetric; for transitivity, we can exploit the flexibility to choose the subcategory in the definition of the relation.
Thus, we have an equivalence relation on $\mathcal{D}_\rightsquigarrow$.
Using Scott's trick, or the axiom of global choice if you prefer, define $\mathcal{D}_1$ to be the class of representatives for this equivalence relation.
Define $\mathcal{D}_2$ to be the class of composable pairs of elements of $\mathcal{D}_1$, etc.
The point is that localisation is a finitary construction on categories, meaning that the functor $(\mathcal{C}, \mathcal{W}) \mapsto \mathcal{C} [\mathcal{W}^{-1}]$ preserves filtered colimits.
Every category – morally, at least, even if our logical framework makes it extremely awkward to say exactly what we mean – is the filtered colimit of all its small subcategories.
In the definition of the equivalence relation on $\mathcal{D}_\rightsquigarrow$ above, I exploited this fact to avoid giving an explicit construction (which can be done – see the papers of Dwyer and Kan!) and reduce to the case of localisation of small categories.
In the case of the derived category of an abelian category, there is a somewhat simpler description, owing to the fact that the category of chain complexes is a category of fibrant objects: morphisms in the homotopy category of a category of fibrant objects can always be represented as $f \circ w^{-1}$ for a fibration $f$ and a trivial fibration $w$ (i.e. a cocycle in the sense of Jardine), and composition can be represented by composition of spans.
But as everyone has already said, this is not enough to make the class of morphisms between two objects small enough to be a set.
Personally, I find it quite awkward not being able to form collections of classes.
It is possible to iterate the ZFC to NBG construction but it seems not well studied, let alone actually used.
It is much easier to work with one or more Grothendieck universes in an ambient set theory.
One interesting thing is that set theory in a Grothendieck universe can be stronger than the ambient set theory – for example, even in Mac Lane set theory, a Grothendieck universe is a model of full ZFC (and more; for example, it also satisfies the axiom of global choice, $\textrm{Con} (\textrm{ZFC})$, etc.).
