I'm trying to book a reward flight from LAX to LHR, but no current availability. I'm told seats do become available nearer departure but wonder how late I should leave it before booking a paid seat?
That is a risk only you can decided. The danger is, if you wait too long, then all the lower fare buckets will sell out leaving you with only the very expensive higher fare buckets left to pay.
Once rewards are initially all gone, then it is usually in the days just before the flight that some more might be released, but there is really no rule as to how and when they decide to release them.