Welsh joined Charlton Athletic in 1935, costing a fee of £3,250 from Torquay United. An extremely versatile player, he could play equally well at inside left, centre forward, centre half or left half. He soon became captain and skippered Charlton to consecutive promotions from the Third Division to the First Division. In 1938 he gained the first of three caps for England against Germany. He later played for the national team against Switzerland and Romania, scoring once against the latter in 1939. Welsh did make 8 unofficial war time International Caps for England between 1940 and 1941, scoring a total of 9 goals, of which he scored all four goals in an impressive 4-0 victory against Wales in 1940.
During the Second World War he made guest appearances for Charlton and Liverpool. He scored 100 goals in 119 League games for Charlton and 43 goals in 40 games for Liverpool. Welsh also led Charlton to two War Cup finals in 1943 and 1944, winning in 1944 by beating Chelsea. He also made nine appearances for England in the semi-official wartime matches, scoring twelve goals.
He returned playing for Charlton at the end of the war. He captained the side to FA Cup finals in 1946 and 1947, winning the latter game against Burnley. He left the club in November 1947 to begin his management career.
With his playing days over, Welsh started his managerial career at Brighton & Hove Albion in November 1947 aged 36 years old. His managerial career did not start well – Brighton finished bottom of the Third Division (South) at the end of the 1947–48 season. Brighton recovered to finish 6th and 8th in the next two years. Following this he joined Liverpool as manager on 5 March 1951 as a result of George Kay's decision to step down through ill-health.
In 1954 Welsh became the first manager to guide Liverpool into relegation for over 50 years. Unfortunately, that day also saw Everton promoted back to the First Division. He had almost accomplished that feat the season before finishing 17th, and only a scrappy win over Chelsea on the last day of the season kept Liverpool up. After this relegation Liverpool would stay in the Second Division for eight more seasons. Liverpool came near to being promoted again in 1955–56, but Welsh would not be given another chance and was sacked as Liverpool manager in 1956. He became the first of the only two Liverpool managers to ever be sacked by the club.
After leaving Liverpool, Welsh became a publican for a while in the West Country before the lure of football brought him back into club management again, this time at Bournemouth in the newly formed and non-regional Third Division just before the start of the 1958–59 season. After two average seasons (they finished 12th and 10th), he was dismissed in February 1961 following a string of poor results. In 1962 he worked for the London County Council Education Department as a Football Coach/Teacher at Tollington Park School in North London. He left the LCC to manage non-League Wycombe Wanderers for a brief spell between July 1963 and November 1964 before returning to the club where he had enjoyed such success as a player, Charlton Athletic, to become a member of their administrative staff. He died in 1990, aged 78. In 2013 he was inducted into the Charlton Athletic Hall of Fame.