The correct sentence would be: "my time has been spent catching up with..." You'll want to use "catching" (i.e. the Present Perfect Continuous) and not "caught" (i.e. the Present Perfect) because the PPC focuses on the length of time of an action, whereas the PP is generally neutral regarding time. Also, the PPC is used when the result comes from the action itself. It doesn't matter if the whole action is finished or not.

There may be regional variations. I wouldn't say "my time is caught up" There are many usages in the US "I've been caught up with work" (I've been unavailable because of work) I'm caught up in something that I don't understand. (I'm involved (perhaps involuntarily) in something that I don't understand) I'm trying to catch up on my sleep" (I'm behind and trying to change that) I'm trying to get caught up. (I'm behind and trying to change that) I'm caught up. (I'm not behind any more)