The one-point compactification of the discrete real line is compact and Lindelöf but is neither first countable nor separable

Give $D=\mathbb R$ the discrete topology and form $D^*=D\cup\{\infty\}$. The one-point compactification theorem makes $D^*$ compact, hence Lindelöf, and every point of $D$ remains isolated.

A neighbourhood of $\infty$ has finite complement in $D$, since compact subsets of a discrete space are finite. If $(N_n)$ were a countable local base at $\infty$, put $F_n=D\setminus N_n$. For every $x\in D$, the neighbourhood $D^*\setminus\{x\}$ would contain some $N_n$, so $D=\bigcup_nF_n$. Each finite subset of $\mathbb R$ has a canonical increasing enumeration; these enumerations and countability of $\mathbb N\times\mathbb N$ make the displayed union countable, contradicting uncountability of $\mathbb R$. Thus $D^*$ is not first countable. Finally every dense subset must meet the open singleton $\{x\}$ for every $x\in D$, so it contains all of $D$ and cannot be countable; hence $D^*$ is not separable.