Yes, no fuss, just include him on an outing or something and introduce him as “John” or whatever his name is.
The important thing is not to force things, no “Aren’t you going to give John a kiss too?!”, or insisting he gets a grandpa name.
My grandma’s boyfriend was “Uncle Firstname” same situation for my childhood best friend. That was nearly 50 years ago and I think just the first name would probably be more normal now.
Also your grandchildren would probably prefer to still get some time with just you and not have you always come as a package.
Children are very adaptable, they probably won’t bat an eyelid.