I'm quite conflicted over this. When I saw it I actually wondered what the hell it was about and finally thought it must be racist because of the chimp. However, it didn't stand out as glaringly obvious to me because both the parents look like white British. It would definitely have been stood out as outrageously racist if one parent had been black.
When I read Danny Baker's explanation after he'd been sacked and he explained it was a gag about "a posh chimp" and that he would have done the same had it been Boris Johnson (meaning any other posh person) leaving the hospital with a new baby, then I can see what he means.
He has explained it and said he deleted the tweet when the racist connotations were pointed out to him but I think he should have apologised wholeheartedly for his stupidity and lack of foresight.
I tend to believe him , he didn't see it as racist because he doesn't see race, or as he says his mind isn't diseased. He's just a bit of a twat! Oh well, he's lost his job and getting a kicking in the press and social media so he's got his punishment.