No, it need not be proper.
Indeed, let $T$ be a real tree consisting of one vertex $v$ with a sequence $(e_n)$ of edges of length $1/n$. Then $T$ is compact.
But the tangent cone of $T$ at $v$ is a real tree in which there are, for every $r>0$, infinitely many points at distance $r$ from $v$, pairwise at distance $2r$. So it is not proper, and not even locally compact.