I think the front left shock absorber on my 2008 Fiesta has broken. When I sit on the wing and get off the wing doesn't immediately stop like the other side. Is this a fail at MOT test? 

Yes

Technically yes but probably not in reality.  They're more interested in leaks than the rebound rate.  Some cars do just have bouncy shocks as standard so unless they specifically bounce both sides they may not notice in this case.

5.3.2. Shock absorbers

You should reject a missing shock absorber only if they were fitted as standard.

A shock absorber must be rejected if negligible damping effect becomes evident at any point during the inspection.

Slight seepage causing a film of fluid on a shock absorber is not a reason for rejection.

Defect	Category
(a) A shock absorber: (i) insecurely attached to chassis or axle (ii) missing or likely to become detached	Major Dangerous
(b) A shock absorber damaged to the extent that it does not function or showing signs of severe leakage	Major
(c) A shock absorber bush excessively worn	Major
(d) A shock absorber which has negligible damping effect	Major