Login onto united.com, verify that
"My Profile"
->"My Travel Preferences"
->"Display Preferences"
->"Availability Format" is set to "an expert"

Then search for "standard award" milage award booking for the specific flight you are interested in learning about "current" upgrade inventory availability.

When the list of flight options comes up, click on "Fight info" for the flight of interest. You are looking for the NF NC items on the list.
NC# is the number of clearable upgrades to business class currently available.
NF# is the number of clearable upgrades to first class currently available.

If the upgrade you want is currently not 0, then your upgrade should clear right away if you are inside the window for your status, or it should clear at anytime if you are using miles, CR1, or SWU.

If the upgrade you want is currently 0, then your upgrade will NOT clear right away, and you will be placed on the waitlist. The probability of your upgrade clearing at this point is impossible to know, and you just have to wait and see.

If NC(business) or NF(first) seats become available after your request, I think that they will clear as they become available based on status/fare ordering rules.

I have used this feature on united.com and I assume that many FT folks know about this, but I was reading FT a while before I learned about it.

Moderator's Note - Older, related thread may be found here.