Blencowe began racing in 1999 in the 750 Motor Club's Stock Hatch Series, building a competitive Peugeot 205 GTI himself. He won one race and finished fourth in the championship that year. In 2000 he moved to the Ford Fiesta Zetec Championship, finishing fifth in the main series and winning the Ford Fiesta Winter Series championship. He contested the Fiesta series for a second season, again finishing fourth.
For 2002 Blencowe stepped up to the BTCC with Gary Ayles (GA) Motorsport, driving an Alfa Romeo 156 in the Production class. His debut weekend was disrupted: the team's second driver collected him in race one and another competitor took him out while he was running third in race two. Over the full season he collected four Production class podiums, his best a second place at Thruxton, and finished seventh in the Production class standings.
In 2003 Blencowe joined Barwell Motorsport, switching to a Honda Civic Type-R. Despite taking only one Production class win, he accumulated 18 podiums across the season with just a single retirement, and his consistency earned him second place in the BTCC Production class championship.
With sponsorship difficult to find for 2004, Blencowe competed in the Britcar Endurance series with Barwell Motorsport in a Honda Civic Type-R. He entered four rounds and took two class wins, consistently proving the fastest of the two- or three-driver crews sharing the car.
From 2005 Blencowe moved to the SEAT Cupra Championship, driving for Triple R. He recorded four third-place finishes that year and placed fifth in the standings. In 2006 he was runner-up in the championship, recording one win and ten podiums including a lap record at Knockhill. His performances attracted attention from SEAT's factory operation and at the end of 2006 he was invited to test for the Seat factory team at Rockingham.
Blencowe remained at Triple R for 2007, driving the new SEAT León. He won two races but the season ended prematurely after a serious accident at Snetterton caused by the pole-sitter left him injured.
In 2008 Blencowe was engaged by Synchro Motorsport, running Honda's factory Swindon operation, for two races. That programme ended when an oil leak caused the car to catch fire at Snetterton. Following his retirement from front-line competition Blencowe worked as a professional driver for Lamborghini on a contract basis and established himself as a one-to-one driver coach operating across the United Kingdom.
In 2023 Blencowe purchased a 2018 Gen-1 TCR Cupra and re-entered competition in the 750 Motor Club's Club Enduro Championship in 2024, running under the Alan Blencowe Racing (ABR) banner. The team treated 2024 as a development year, securing a class win at the final round. For 2025 ABR entered both the Roadsports and Club Enduro Championships run by the 750 Motor Club, aiming to build on that result and accumulate more podiums.