As a former Australian Army Regular and Reserve Soldier with 14 years service who was for a couple of years also an Infantry Platoon Sergeant. I can say from experience that soldiers do sometimes fire their weapons like rifles, machine guns, pistols, and sub-machine guns from the hip and also train to do so with acceptable accuracy at short ranges. So firing from the hip outside of Hollywood does occur even with professionals.
I have not had any experience with the SA80 rifle that said however, I am not unfamiliar with infantry rifles with my experience including usage and instructing on the L1A1 7.62mm SLR, M16 and some variants 5.56mm and the F88 Steyr 5.56mm amongst a considerable variety of other infantry weapons.
I have never had any problems with balance firing weapons from the hip, bearing in mind that one uses small bursts like 3-5 rds for example if firing at an automatic rate to assist with accuracy and control.
One thing worth noting however regarding balance is that although the F88 Steyr is a bullpup rifle like the SA80, its balance is fine normally. It does change becoming slightly more forward heavy when one attaches items like the NAD Night Aiming Device and a Surefire light for example. This change of balance also occurs with a fighting knife (bayonet) fitted or with the M203 40mm GL being attached for example. That is why soldiers train with their weapons in various conditions with both Blank and Live ammunition in order to better apply these weapons regardless of condition.