Beverly states with confidence that Shinzon is 25 years old. Since the film takes place in 2379, this means that Picard's clone would have been created when he was still in command of the USS Stargazer in 2354, some ten years before he gained command of the Enterprise.
In-universe there are two possible explanations for this discrepancy;
Picard was already marked for greatness. His command of the Stargazer was exemplary and he was known as a military genius who'd invented his own unbeatable maneuver.
Tasha Yar, thrown back in time in TNG : Yesterday's Enterprise told the Romulans that Picard would one day be in command of the flagship of the Federation.
BEVERLY : There's no doubt, Captain. Right
down to your regressive strain of
Shalaft's Syndrome. He's a clone.
A beat as the confirmation sinks in.
PICARD : When was he... created?
BEVERLY : About twenty-five years ago. They probably used a hair
follicle or skin cell.
As to how they planned to replace Picard despite the age difference, Beverly works it out:
BEVERLY : The more I studied his DNA the more confusing it got. Finally I could only come to one conclusion... Shinzon was created
with temporal RNA sequencing. He was designed so that at a certain
point his aging process could be accelerated to reach your age more
quickly, so he could replace you.
PICARD : But the Romulans abandoned the plan...
BEVERLY : As a result the temporal sequencing was never activated. Remember, he was supposed to replace you at nearly your
current age. He was engineered to skip thirty years of life. But
since the RNA sequencing was never activated, his cellular structure
has started to break down. He's dying.
Elsewhere in the Star Trek canon, according to "Death in Winter", the harvesting of Picard's DNA supposedly took place in 2348 at the wedding of Jack and Beverly Crusher. To add insult to injury, Picard wasn't the only Starfleet officer that was cloned :
But then, who would suspect him of being a surgically altered Romulan
spy—an agent dispatched across the deceptively quiet Neutral Zone in
support of a program only the praetor, in his brilliance, could have
A plan to grow clones from the genetic material of Starfleet’s most
prominent captains and, at some opportune juncture years or even
decades hence, replace them with their secret progeny. Brilliant was
probably an under-statement.
It wasn’t difficult for the Romulan to make off with his prize.
Picard’s gaze had already drifted in the bride’s direction again. And
in the process, Manathas had been forgotten. Making certain no one was
watching, he emptied what was left of Picard’s champagne into another
glass—one that had earlier been the property of the bride—careful not
to disturb the smudge where the captain’s mouth had come in contact
with the transparent rim.