Joking about some accents is very tricky. People often find the Indian English accent very amusing and imitate it to get laughs (think Peter Sellers in The Party) but that does not always go down well. I had a Swedish student once who, when he spoke English, automatically mirrored the accent of the person he was speaking to. He had almost been beaten up a few times and said that Chinese and Hindu people had taken extreme offence because they thought he was laughing at them. No matter what I tried I couldn’t train him to stop doing this. I often wonder where he is now and if he still does it.
So yes, accents can be a sensitive topic. Not just in places like the UK where they denote social class, but also in countries like South Africa where imitating someone from another race can really offend.
Not so with Pieter Dirk Uys and Bishop Desmond Tutu. Pieter Dirk Uys is a white Afrikaner who defied the Apartheid government by imitating all the leaders of that time and lampooning both them and the apartheid regime. His most famous character is Evita Bezuidenhout (almost like Dame Edna Everage). You can see how much the Tut loves this skit. His laughter always makes me happy!