Welsh was born in Torrance, California, in Los Angeles County to Pat and Kathy Welsh being the eldest brother of Jack and Henry Welsh. At age five, he started playing basketball. He attended Loyola High School in Los Angeles. As a freshman, Welsh played basketball on the junior varsity team, where he was ranked eighth on the team's depth chart. He was promoted to the varsity team as a sophomore, moving up the depth chart to seventh. He did not become a starter until his junior year, by which time he had grown to. Still, he was only considered a prospect for mid-major colleges at the time, or perhaps a redshirt at a Pac-12 Conference school. The following summer, Welsh performed well with the Los Angeles Rockfish, the longest ongoing high-school all-star program in Southern California. He emerged as one of the top centers in California as a senior, and was selected for the 2014 McDonald's All-American Game, a rarity for a player who did not receive national recognition as a junior. He was rated as a four-star recruit.
College career
Freshman
In November 2013, Welsh committed to play at UCLA over fellow Pac-12 schools California and Stanford. As a freshman with the Bruins in 2014–15, he spent most of the season as a backup to junior Tony Parker. Welsh played in all 36 games, starting in three, and averaged 3.8 points and 3.8 rebounds per game. He led the team in blocks, averaging 1.1 per game as a reserve. He became the Bruins' first 7-foot player since Ryan Hollins in 2005–06. In the season opener against Montana State, Welsh scored 14 points in 13 minutes of play. He had six rebounds and four blocked shots in 22 minutes in an upset victory over SMU in UCLA's opening game of the 2015 NCAA Tournament. Playing the final minutes of the game in place of Parker, he made a key block with 34 seconds remaining and UCLA down by four.
Sophomore
Having played on the U.S. under-19 national team over the summer, Welsh's footwork and fundamentals improved. With Kevon Looney having moved on to the National Basketball Association, Welsh was promoted to UCLA's starting lineup at center, while Parker moved to forward. In the 2015–16 season opener, he had his first collegiate double-double with 12 points, 10 rebounds and a career-high five blocks in an 84–81 overtime loss to Monmouth. The blocks were the most by a Bruin since Travis Wear's five in 2012. In the following game, Welsh scored a career-high 22 on 10-for-12 shooting in an 88–83 win over Cal Poly. On December 3, against No. 1 Kentucky, he registered game highs of 21 points and 11 rebounds and outplayed highly touted Wildcats freshman Skal Labissiere in an 87–77 upset win, the Bruins' first victory over a No. 1-ranked team since 2003. On January 9, 2016, he had a season-high 16 rebounds in an 81–74 win over Arizona State, which helped UCLA avoid falling to 1–3 in the Pac-12 for the first time in almost 20 years. After UCLA lost three of four games in mid-February, Welsh became a reserve after offering to coaches that Parker could take his place in the starting lineup; the senior Parker had been benched in favor of a quicker Jonah Bolden. The Bruins lost their final five games to finish with a 15–17 record. Welsh ended the season with an 11.2 scoring average, and led UCLA in field goal percentage, rebounding and blocked shots. He ranked seventh in the Pac-12 in rebounding, second in offensive rebounds per game, and third in field goal percentage.
Junior
UCLA began 2016–17 ranked No. 2 after starting 13–0, the first time they were undefeated in non-conference play since they won a national championship in 1994–95. Welsh was leading the Bruins in rebounding again, but had missed the last four games with a bruised right knee. He returned in the conference opener, but the Bruins suffered their first defeat in an 89–87 loss to No. 21 Oregon despite Welsh's 20 points, 10 rebounds and three blocks. On February 19, 2017, he scored 16 points and had a season-high 16 rebounds in a 102–70 win over their crosstown rivals, USC, ending a four-game losing streak against the Trojans. On February 25, Welsh scored 12 of his 14 points in the second half to lead No. 5 UCLA to a 77–72 win over No. 4 Arizona, which also snapped their conference rivals' 21-game home winning streak. In the Bruins' regular-season finale, he had 13 points and 11 rebounds and also made his first career three-point field goal in a 77–68 win over Washington State. The Bruins ended the season 31–5 after losing in the Sweet 16 of the 2017 NCAA Tournament to Kentucky. For the season, Welsh averaged 10.8 points along with team highs of 8.7 rebounds and 1.3 blocks per game, and he received an honorable mention for the All-Pac-12 team. Although he was not listed in most mock drafts, he declared for the 2017 NBA draft. After attending the NBA Draft Combine but not hiring an agent, he withdrew his name from the draft to retain his college eligibility.
Senior
Over the summer, Welsh worked on his three-point shooting based on feedback from his NBA workout with the Oklahoma City Thunder. He was the Bruins' lone returning starter in 2017-18. On December 9, 2017, he tied a career high with 22 points and added 10 rebounds in a 78–69 overtime loss to Michigan. On January 4, 2018, Welsh required three stitches after being bloodied by a hit to his nose. He was forced him to wear a protective mask after halftime, which affected his shooting as UCLA lost 107–99 in double overtime. He continued wearing a mask the following game, but adjusted for 19 points, 14 rebounds, and a career-high four three-pointers in a 107–84 win over Cal. On February 15, he played his first game without a mask since January and logged a career-high 17 rebounds along with 15 points in a 75–68 victory over Oregon State. In UCLA's Pac-12 Tournament opener, Welsh had 18 points, 11 rebounds, and tied his career-high of four three-pointers in an 88–77 win over Stanford. The Bruins qualified for the 2018 NCAA Tournament, but lost 65–58 to St. Bonaventure in the First Four for the Bonnies' first tournament win in 48 years. Welsh was limited to just two points, but had 15 rebounds to pass David Greenwood as the Bruins' third all-time leading rebounder, behind only Kareem Abdul-Jabbar and Bill Walton. Welsh completed the season averaging 12.6 points and 10.8 rebounds. He made 40.2 percent from 3-point range after attempting just one in his first three seasons. One of just two players along with Arizona's Deandre Ayton to average a double-double in the Pac-12, he earned second-team All-Pac-12 honors. The United States Basketball Writers Association voted him to their All-District IX Team, and he was a finalist for the Kareem Abdul-Jabbar Award, which recognizes the top center in Division Imen's college basketball.
Professional career
Denver Nuggets (2018–2019)
In the 2018 NBA draft, Welsh was selected in the second round with the 58th overall pick by the Denver Nuggets. On July 19, 2018, Welsh was signed by the Nuggets to a two-way contract. Because the Nuggets were one of only three NBA teams to not have a NBA G League affiliate, Welsh was assigned to Washington Wizards affiliate Capital City Go-Go. Nuggets assistant coach Wes Unseld Jr. was impressed early by Welsh's constant talking—calling out coverages and directing teammates on the court—in spite of his youth and inexperience. Welsh made his NBA debut on October 17, 2018 against the Los Angeles Clippers, seeing under one minute of game action and scoring no points. On January 1, 2019, the Nuggets transferred Welsh from Capital City to the Iowa Wolves. On July 30, 2019, Welsh was waived by the Nuggets.
Greensboro Swarm (2019–present)
On August 9, 2019, Welsh signed with the Charlotte Hornets. He was waived during training camp in October, but joined their G League affililate, the Greensboro Swarm, later in the month as an affiliate player. On January 17, 2020, he posted 22 points, 15 rebounds and four assists in a loss to the Fort Wayne Mad Ants. Welsh averaged 10.9 points, 9.0 rebounds and 1.7 assists per game.
National team career
Welsh tried out for the United States national team in the 2014 FIBA Americas Under-18 Championship, but was one of the final three roster cuts. The following year after his first season with UCLA, he was one of 12 players selected by the U.S. to play at the 2015 FIBA Under-19 World Championship in Greece. He was not considered a top contender when tryouts began, but he made an impression with his rebounding and perimeter play. The Americans won the gold medal after a 79–71 win over Croatia. Welsh played in all seven games, averaging 2.7 points and 3.3 rebounds in 8.9 minutes.
