Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?

The real numbers are a complete densely ordered set without endpoints. That is, there is no minimum, no maximum, between every two points there is a third, and every set which has an upper bound has a least upper bound.

Theorem: Every countable dense order without endpoints is order-isomorphic to the rational numbers.

Since the rational numbers are not order complete, the real numbers are not order-isomorphic to the rationals. Therefore the real numbers cannot be countable.

The real numbers are a perfect set, and all perfect sets are uncountable. In particular, this gives a proof of the uncountability of real numbers that does not reference decimal expansions.

Not all proofs of uncountability of the reals involve diagonalization. In fact, one can prove without diagonalization that $\mathbb R$ and $\mathcal P(\mathbb N)$ have the same size, and then give a diagonalization-free proof that, for any $X$, its power set $\mathcal P(X)$ has size strictly larger. This was first noticed by Zermelo. The details can be found in this MO answer.

Briefly: You prove that if $f:\mathcal P(X)\to X$, then $f$ is not injective, by explicitly exhibiting a pair $A\ne B$ of subsets of $X$ with $f(A)=f(B)$. Zermelo's approach uses well-orderings. You find $A,B$ by using transfinite recursion, to define an injective sequence $\langle a_\alpha\mid \alpha<\tau\rangle$ of elements of $X$ such that for all $\beta<\tau$ we have $f(\{a_\alpha\mid \alpha<\beta\})=a_\beta$, but $f(\{a_\alpha\mid \alpha<\tau\})=a_\gamma$ for some $\gamma<\tau$.