Good news, they are both legal.
BREAKING NEWS!!!!
Today Monday 26th June 2017 we have officially been informed that the Steyr LP50 is indeed self indexing and not self loading as defined by section 57(2A) of the Firearms Act 1968. Therefore does not fall within any category of Section 5 or Section 1.

As the Steyr Hunting 5 Auto is the same mechanism and sub 12 foot pounds muzzle energy this also does not fall in the Section 5 category of self loading.

Sales, service and repair are now fully resumed