That’s like me, never won anything playing basketball growing up but when I turned 40 and started a team in the local over 30’s comp - undefeated two seasons in a row. Then I retired, too many young blokes.
He struggled in the first part of his pro career, but he won the Heisman at Stanford, and he was the first overall pick of the 1971 draft, so I don’t think he fits the OP’s criteria. Expectations were very high for him (Tommy Prothro called him “the best pro quarterback prospect I’ve ever seen”), but it took him 9 years, 2 team changes, and 2 years on the bench before having several very good years with the Raiders.
Ben Wallace, undrafted, played sparingly in Washington, a little bit in Orlando, then became an All Star with the Pistons in year 5 or 6.
My brain just threw up.
My contribution is in the special longevity category: Jamie Moyer. On the date of his 30th birthday he was a journeyman lefty with 34 career wins as a starting pitcher and wasn’t really notable for anything other than letting the other team hit the ball over the fence. Since his 30th birthday he has racked up 229 wins and counting, including an about-eight year peak from ages 32-40 where he was garnering a few Cy Young votes and was a legitimately outstanding pitcher, though never an elite one. His ERA+, where 100 is the league average, was 87 before he turned 30, was 121 during his peak years, and now stands at 109 in the 17 (!) seasons he’s pitched since turning 30.
As I understand the OP, expectations are irrelevant. And Heisman Trophies mean nothing in the pros. I sometimes think they’re negative predictors; show me a Heisman Trophy winner, as often as not I’ll show you an NFL bust.
And the #1 overall draft pick status doesn’t really matter, either. Consider Jamarcus Russell. If he finds a new team and in, say, three years becomes a top 10 QB in the league, wouldn’t he qualify?
How about Flo Jo? She was certainly a good runner early but later seemed to be really at a level she hadn’t achieved.
There are several similar stories like this on the Senior Tour. Gil Morgan, Miller Barber, and Bruce Fleisher were journeyman players on the PGATour and went on to be dominant players on the Senior tour. There were players who barely played the PGATour who went on to successful careers on the Senior Tour like Tom Wargo and Allen Doyle.
He was good all along, but even the best QB can do nothing if he has no blocking and can’t run away.
Another golfer to mention is Lee Trevino, who was a club pro who made more money betting on trick shots than on playing until trying the tour in his late twenties.
Jersey Joe Walcott was an average boxer in his early career. He won a lot of bouts, but all boxers pad their records with creampuffs; when he fought top contenders, he lost. But he turned it around in 1945 when he was 31 and started rising in the ranks. He finally got his chance at the title when he was 33, then lost four times (twice to Joe Louis and twice to Ezzard Charles) before winning the championship at the age of 37, the oldest (at that time time) to win it.
In a sad note, AP is reporting that Chi Chi was tied up and robbed of $500K in money and possessions at his apartment last night.
But what other golfer was immortalized by Les Nessman on WKRP? 
I think they’re two different kinds of stories.
Kurt Warner and Rich Gannon are examples of guys who came out of college relatively unheralded, and scuffled around for some time, before suddenly playing at an All-Pro level. Up until they had their breakouts, they were considered journeymen, at best. To me, that’s “suddenly becoming outstanding.”
Jim Plunkett (or, for another example, Desmond Howard) is a guy who came out of college and entered the NFL with much attention, with everyone expecting them to perform at a high level. At first, they didn’t. Eventually, they did find a niche (or, at least, had a short period of playing at a high level). To me, that’s not “suddenly becoming outstanding”, because people always thought he was going to be outstanding…it’s more an example of “an underachiever who starts playing at his ability level.”
JaMarcus Russell could, in theory, be the second kind of story, if he manages to get another job, and get his act together. He obviously will never be the first kind.