Wrong way round Ed, the 416 was H&K ripping the guts from an L85 and G36 and putting them into an M4-chassis.
In turn, the system from those came from Armalite/Sterling in the mid 1960s.
BAE / H&K's rebuild of the L85 - wasn't a rebuild. It's literally the same parts as the A1. Only the parts are made to better tolerances. H&K did not redesign anything. You open it up, and the internals will look exactly the same as the parts from Sterling's AR-180 - which is what Enfield based the SA80's gas parts on. Only now, the parts have H&K stamped on them. (And are painted a dull red when they leave the factory)
The only change, was what the parts (bolt, cocking handle, recoil/buffer springs, hammer, firing pin, extractor-assembly) were made from, and how they were made. When the SA80 was originally built, it was made by RSAF-Enfield staff who were (due to the impending privatisation and selling of RSAF) about to be made redundant and did not do their usual good job. Corners were cut, money was saved at the expense of doing a good job.
H&K were used because in 2001, they were (and still are) the only firearms manufacturer left in the UK, with operational factories. Had Sterling not gone bankrupt in the mid 1990s, they would have done the same job. Hell, had BAE not closed the RSAF after they bought it, they'd have probably done the job anyway.