I'd also like to add that Airsoft doesn't count as an air weapon because Airsofters managed to get an exclusion in the law. Paintball didn't need to get this exclusion because projectiles that burst on impact can't be a weapon.
Airsoft were able to use full auto prior to the laws exclusion. The airsoft exemption relates to the Violent Crime Reduction Act. (VCRA - sometimes referred to as VCRB in relation to the bill proposed before the act came into force)
Full auto is not mentioned in the VCRA. If falls within existing firearms laws.
The law can be considered a grey area and requires a court case to decide. This is one of the principles of law in the UK. Even when a case goes one way other cases haev to be related to that case, and can also be referred to higher levels. What matters prior to that is Home Office advice to the police.
UKPSF/Steve Bulls published information go into that, and there is more detail to be found in the UKPSF pages of when he demonsrated fire modes.
In summary full auto and burst modes are considered not illegal, but unusable on sites due to insurance. Therefore it is legal to posess an auto capable marker. Certain markers come with a UK board, but some US boards have been sold on original markers in the UK (e.g. the first batch of SP8s in the UK) and aftermarket boards are legal.