Merriam-Webster: to return to normal health or strength after being sick, injured, etc.
My first thought was @ColinFine's answer, grieving. It emphasizes a necessary condition for getting over the loss of a loved one. But as @QPaysTaxes points out it's not sufficient. It could be used to describe hopelessness. "He can't stop grieving." I also like closure. However that could apply to someone who never mourns at all, if they come to an "immediate closure."
Recovery comes close, but the emphasis is different.
Recovery = closure + implied grieving
Recuperation = grieving + implied closure
I would say the OP "process of getting over" emphasizes the grieving.
Recuperate usually applies to physical or financial recovery. But googling "recuperate|recuperating from the death of" turns up over 200,000 uses. So I suggest this use fits within the "etc." part of the above definition, and health can include emotional health.
recuperating from the death of his wife and for the most part he appears to have a good support system
Maggie has recently been recuperating from the death of her last family member.
a peer support program that helps struggling kids recuperate from the death of a love one.