How long does it take for non-indigenous plants to start spreading uncontrollably?

Some species of plants can spread in a few weeks, others may take years. I suggest you contact your local botanical garden and speaking to them. Non indigenous plants should be routed out, as they do not provide any adequate habitat for indigenous species of insects and animals. The my be adapted for harsher conditions, and now grow in less harsh conditions, causing them to flourish and take over.