Whether or not you get the bonus will most likely depend on what class your ticket is booked in.

Usually first class upgrades are booked in a different class than purchased first class tickets. For instance, on Delta, first class upgrades are G and V class (which actually get shown as the original fare class plus the G or V in some places that can handle two digit fare codes, for instance an H fare that is upgraded to V will be HV). Purchased first class tickets on Delta are A and F on the other hand. It is A and F class that get the bonuses, not the free upgrades.

However, as some people have pointed out, sometimes free upgrades are given using regular first class codes, for whatever reason, and so if you have a fare code showing a paying ticket, you should get the perks of a paying ticket.

Note that this can work in coach too. I was flying ATL-SAN last year on an award ticket, and the flight was cancelled, so I got rebooked on an ATL-SNA flight, and they rebooked me in Y, meaning I got milage and bonusus and credit for the flight (a nice little bonus), and in that case the delay was negligible (maybe 30 minutes extra to get to my final destination).