Greg spent ten years at Norwich joining as an apprentice in March 1975 and transferring to Coventry City in July 1985. Originally a striker, he was converted to a full-back by reserve team coach Mel Machin. Although his league debut and first senior goal came whilst on loan to Torquay United in 1977, Greg accumulated 206 appearances for the Norwich first team with only seven being as substitute. 8 goals came in that period but there were many more assists as Greg played as an attacking fullback He won Norwich's 1981/82 Player-of-the-Year award but his greatest footballing honour was winning the 1987 FA Cup Final with Coventry in a 3-2 victory over Tottenham Hotspur. 146 games for the Sky Blues in five years were followed by a transfer to fellow Midlanders Birmingham City.

However his spell at St Andrews lasted just 17 games before joining former Coventry manger John Sillett at Hereford as player-coach. The constant struggle against relegation to the Vauxhall Conference saw Sillett lose his job which was filled by Downs in April 1993. He has since left Hereford to play amateur football for Raunds Town, Redditch Town, Kettering Town, Merthyr Tydfil, Worcester City and Forest Green. Mike Davage added in his article "Canaries Winging it Everywhere" in the 1997/98 Norwich City handbook that Greg had played for Bridgnorth Town and Malvern Town in between leaving Forest Green and joining Merthyr Tydfil.

Teamtalk revealed on 30th September that Greg, who had been playing for Merthyr Tydfil in the GM Vauxhall Conference, had returned to his roots to join Jewson Eastern Counties League side Great Yarmouth Town, managed by former teammate Dale Gordon. To support him and his family, Greg is a policeman in the Norfolk seaside town of Gorleston. Greg parted company with Great Yarmouth in the summer of 2001 and joined Anglian Combination side Mulbarton United for whom his son Gareth also played.

According to a conversation we had at the Norwich City centenary dinner (8th September 2002), Greg has retired from playing and his policing career has taken him to Norwich's Bethell Street station. However the lure of the game brought him out of retirement for Blofield but he was sent off in the Norfolk Senior Cup in a 5-0 defeat at home to Great Yarmouth.

In February 2003, Greg joined Dereham Town as number 2 to their new manager Jimmy Landamore. After a poor start to the 2003/04 campaign, Landamore and Downs were sacked in mid-September 2003. By early November, Greg had signed for North Walsham Town.

In March 2006, Greg offered his services to Wroxham chairman Tom Jarrett and joined the Trafford Park management team as assistant to Wroxham team boss Damian Hilton. Hilton said: “This is an ideal appointment for all concerned. I felt very lonely when we lost to Leiston earlier in the month and it would have helped then to have had a partner to lean on. Aside from that I may have to play from time to time.” Downs said: “I have wanted to get back into Ridgeons League football in an administrative capacity, offered Wroxham my services and was delighted with their immediate interest. I hope Damien and I can strike up a fruitful winning combination.”

Downs and Hilton stepped down from their management roles in November 2007