Is an immersed submanifold second-countable?

Solution 1:

Warm-up
In contrast to the winding of $\mathbb R$, it is impossible to immerse the long line $L$ into the circle $S^1$.
Indeed if you had such an immersion (injective or not) $i:L\to S^1$, it would induce an isomorphism of line bundles $T_L \stackrel {\simeq}{\to} i^*T_S$ on $L$.
Since the tangent bundle of $S^1$ is trivial, the same would be true of $i^*T_S$ and then of $T_L$ .
However Morrow proved that $T_L$ is not trivial.

The general case
A non-second-countable manifold $M$ cannot be immersed into a second countable manifold $N$.
Indeed, if $i:M\to N$ is such an immersion, put a Riemannian structure on the tangent bundle $T(N)$, which is possible since $N$ is paracompact.
The immersion gives rise to an embedding $di:T(M)\hookrightarrow i^*T(N)$ of vector bundles on $M$, which induces a Riemannian structure on $T(M)$. This Riemannian structure in turn endows $M$ with a metric compatible with its topolgy. However a connected metrizable manifold is second-countable . Contradiction.
(I have used that second-countability of a connected manifold is equivalent to metrizability or paracompactness: see here)

Edit: An alternative proof
In case $M$ and $N$ have same dimension one can give a purely topological proof, more in accordance with your wish of avoiding Riemannian structures.
(It suffices to give a riemannless proof of the warm-up) . Here goes:

Recall that the Poincaré-Volterra theorem states that if $i:X\to Y$ is a local homeomorphism with $X$ connected Hausdorff and $Y$ locally compact, locally connected and second-countable, then $X$ is second-countable too.
If $M$ and $N$ have the same dimension , an immersion $i:M\to N$ is a local diffeomorphism and a fortiori a local homeomorphism. The theorem of Poincaré-Volterra thus proves that it is impossible to have both $M$ not second- countable and $N$ second-countable.

[Poincaré-Volterra's theorem is the very last result in the first chapter of Bourbaki's Topology.
It is used, for example, in the proof of Radó's theorem according to which every Riemann surface is second-countable.]