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 infinitely many points at distance $1$ from $v$, pairwise at distance $2$. So it is not proper.