Cyanogen OS 12.1 is currently in testing for the OnePlus One, while an OxygenOS update for Android 5.1 won't land until after the OnePlus 2 arrives.
OnePlus took to its forums to talk about updates coming down the pipeline for its OnePlus One device. On the docket is Cyanogen OS 12.1 (based on Android 5.1), which is currently undergoing testing and Google certification. On the other side of things, an OxygenOS update based on Android 5.1 is in the works, but it won't hit the OnePlus One until sometime after the elusive OnePlus Two arrives:
The entire team is currently heads down working on the release for the upcoming OnePlus 2. We are very excited about the work we've done and can't wait to show you all the new features and improvements we've made. We have a small team which means in order to do our best work, we need to focus. This unfortunately means that we won't have an OTA for the community build of OxygenOS we released back in April for the OnePlus One until after the OnePlus 2 release.
Finally, OnePlus says that a fix is coming for OnePlus One users who are experiencing touchscreen issues with their device. There's no timeline on when it will arrive (other than "soon"), but both OxygenOS and Cyanogen OS will receive the minor update. It's a bit of a mess, but at least OnePlus is being quite transparent about where it is with the updates.
Source: OnePlus Thanks to Adrian for the tip!