Near the end of their qualifying ride at the Chris Hoy Velodrome, Charlie Tanfield came down heavily, meaning they were unable to post a time and will not contest Friday's medal rides.
Tanfield, who also crashed during the Tokyo Olympics, had fallen slightly behind his two team mates Dan Bigham and Ethan Vernon and crashed on the blue line on the inside of the track.
A stretcher was brought out, but Tanfield was able to walk unassisted after receiving treatment.
Had he been able to continue, Britain would have been allowed a second attempt but he was not in shape to ride again.
Three of the four riders have to finish a team pursuit and the fourth member of the British team, Oliver Wood, had already lost pace and dropped out of the race before Tanfield's accident.
"Following his crash in this morning's men's team pursuit qualification, Charlie Tanfield was assessed immediately by the on site medical team before being taken to hospital for further treatment," British Cycling said in a statement.
"We wish Charlie a very speedy recovery and look forward to seeing him back on his bike soon. Unfortunately, the men's team pursuit riders will not progress in the competition."
Britain have a glittering record in track team pursuit and won the world title in France last year when Ethan Hayter, Bigham, Vernon and Wood beat world record holders Italy.
Denmark's quartet of Niklas Larsen, Carl-Frederik Bevort, Frederik Madsen and Lasse Leth topped the qualifying times with 3:46.816, with New Zealand and Italy rounding out the top three.
In Friday's round one Denmark will face Australia, while New Zealand are up against Italy with the winners contesting the gold medal ride on Saturday.
There was better news for Britain in the women's team sprint as they were quickest in qualifying, ahead of Germany. The gold medal will be decided later on Thursday.
American Chloe Dygert was quickest in the individual pursuit qualification and will race against Germany's Francisca Brausse later for the gold medal.
The 11-day UCI World Championships in Glasgow is the first time all the cycling disciplines have come together at one time. The format, featuring track, road, BMX and mountain bike events, as well as para cycling, will be used once every four years, one year before the Olympics.
Glasgow's city centre will host the finish of the men's elite road race on Sunday with the women's elite road race on the final day of competition.
