I know an Ambassador who committed a crime and fled abroad on diplomatic immunity. Will he be able to return to his native country in the future under statute barred or statute of limitations rules? He would have been jailed for five years if he wasn't a Diplomat.
There are several options that the Home country has.
It can choose to waive diplomatic immunity and allow the Host nation (Saudi Arabia in this case) to pursue legal charges in their own system.
The Home nation can choose to pursue charges as though the crime were committed on national soil. Typically once charges are brought the statute of limitations is suspended but this may vary by nation.
The Home nation can choose not to pursue charges. In this case once the statute of limitations expires then the former diplomat would be free at home. And would enjoy the immunity from prosecution for the incident in the Host country as well.
However since the diplomat appears to have fled before charges could be brought, if Saudi Arabia were to gain possession of the former diplomat it could pursue legal action against the diplomat as an international fugitive. The former diplomat could attempt to invoke diplomatic immunity and it would be up to the Home country to either back their former diplomat, or they could choose to deny the claim since the former diplomat fled and they could interpret that as surrendering his diplomatic post and in effect renouncing his diplomatic status.