Nuclear and starburst activity are known to often occur concomitantly. Herschel-SPIRE provides sampling of the far-infrared (FIR) spectral energy distributions (SEDs) of type 1 and type 2 AGN, allowing for the separation between the hot dust (torus) and cold dust (starburst) emission. We study large samples of spectroscopically confirmed type 1 and type 2 AGN lying within the Herschel Multi-tiered Extragalactic Survey (HerMES) fields observed during the science demonstration phase, aiming to understand their FIR colour distributions and constrain their starburst contributions. We find that one third of the spectroscopically confirmed AGN in the HerMES fields have 5σ detections at 250 μm, in agreement with previous (sub)mm AGN studies. Their combined Spitzer-MIPS and Herschel-SPIRE colours (specifically S250/S70 vs S70/S24) quite clearly separate them from the non-AGN, star forming galaxy population, as their 24 μm flux is dominated by the hot torus emission. However, their SPIRE colours alone do not differ from those of non-AGN galaxies. SED fitting shows that all those AGN need a starburst component to fully account for their FIR emission. For objects at z > 2 we find a correlation between the infrared luminosity attributed to the starburst component, LSB, and the AGN accretion luminosity, Lacc, with LSB ∝ Lacc0.35. Type 2 AGN detected at 250 μm show on average higher LSB than type 1 objects but their number is still too low to establish whether this trend indicates stronger star formation activity. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.