Sinclair, only called up on the morning of the game at Trent Bridge after fellow spinner Gudakesh Motie took ill, suffered the painful blow on the fourth evening when a rapid delivery from Wood also flicked the wrist band of his glove before looping to the slip cordon.
His exit for one was part of a West Indies collapse that saw the tourists slump from 61 without loss to 143 all out as England won by 241 runs to take an unassailable 2-0 lead in a three-match series.
West Indies captain Kraigg Brathwaite told a pre-match press conference at Edgbaston on Thursday that Motie would now come back into the side but that the 11 had still to be confirmed as fast bowler Shamar Joseph is suffering from flu.
The uncapped Akeem Jordan, who has been playing locally for Birmingham League side West Bromwich Dartmouth, was called up into the squad on Wednesday after fellow quick Jeremiah Louis -- who did not feature in the first two Tests -- was ruled out with injury.
Now the 29-year-old Jordan, who has taken 67 wickets in first-class matches at 24.10, could make his Test debut at Edgbaston.
"Sinclair is out with a fracture," said Brathwaite. "He got a blow from Mark Wood, "So, Motie's back in. And we have called up Akeem Jordan, so we have have a 12.
"We've named a 12 because Shamar Joseph just has a bit of flu. So, we're waiting on tomorrow (Friday) morning to make the decision on the 11."
Although the West Indies have lost the series, Brathwaite said they would still have plenty of motivation at Edgbaston.
"We have five Tests remaining for the year and, you know, World Test Championship points are still at stake.
"We have a lot to play for and that's my thing to the boys.
"You know, we are still in the running for the Test Championship and obviously we're looking for points as well."
'Got to be on it'
Unlike their innings and 114-run defeat in the series opener at Lord's, the West Indies were competitive for large parts of the match at Trent Bridge and even gained a first-innings lead.
They were well-placed in their second innings before a flurry of wickets cost them dear, Shoaib Bashir doing the bulk of the damage as the 20-year-old off-spinner took 5-41.
"We had a fifty opening stand and then, from there, we lost 10 wickets quite quickly," said Brathwaite.
The 31-year-old opener, out for 48 and 47 at Trent Bridge, added: "It shows that you've got to be on it always, not just for the first three days,
"As soon as you lose that focus and it could be as a bowling unit or a batting unit, you know, and you lose it for an hour, you could be behind the Test match and lose it.
"So we should learn from that and obviously the learning for that is just never taking anything for granted.
"As long as we keep that attitude of fighting every session, I think that will serve us well."
