I found this post because I wondered something related, and I thought I'd post an observation here:
Yes, defining the Borel $\sigma$-algebra goes through trivially. But I think certain fundamental, "nice" theorems don't. For example, theorem 1.8 of big Rudin, page 11, says that if $u,v: X\to\mathbb{R}$ are measurable then so is $u\times v: \mathbb{R}^2$. The argument is as follows.
Given an open set $\Omega\subset\mathbb{R}^2$, the inverse image of any rectangle $(a_1,b_1)\times (a_2,b_2)$ is just $u^{-1}((a_1,b_1))\times v^{-1}((a_2,b_2))$, which is measurable in $X$. Any open $\Omega$ can be written as a countable union of rectangles (here the second countability is used) and so $(u\times v)^{-1}(\Omega)$ is the countable union of these measurable sets, hence is measurable.
This proof goes through precisely the same, replacing $\mathbb R$ with any second countable space. My sense of urgency tells me not to try to construct a counterexample for the long line to say that second countability is needed, but that might be an interesting question.