Oh! Context is everything. Today I found his accent not so bad, but I now understand spoken English a bit and those sounds are also a little bit "my sounds" now.
I found this performance very believable (I think they avoided any rr) and the foreign accent to be indeed a nice touch:
When singing in English Nat King Cole has an English accent that I like and I found that accent easy to understand. I like his voice. He is one of a few dozens of singers that share the quality of being excellent and being also spotted (if spot applies to hearing) within the first two seconds of hearing them, like Frank Sinatra or Karen Carpenter.