"Are We In Danger Of Losing Our Own British Humour, Now That We Live In A Multi-Racial Country?"
No, it would appear not, as evidenced by the past decade of critical and commercial comedy hits from the BBC, such as 'Nighty Night', 'The Thick of It', 'Extras' and 'Life's Too Short'.
In fact, you could argue that comedy is now is way more edgy than it ever has been in the past 30 years.
Whilst the BBC still broadcasts inoffensive material such as 'Miranda', it also shows the popular 'American Dad' and 'Family Guy', both of which are stuffed to the gills with racist, sexist and homophobic jokes and both are hugely popular (because they are, or at least WERE) extremely funny shows.
I don't see anything in the Telegraph report to suggest that this issue is anything to do with race...I suspect, as Gromit has mentioned, that the Beeb is particularly sensitive after 'Sachsgate', where the BBC came under pressure to drop Russell Brand and Jonathan Ross after that infamous broadcast on Radio Two.
So I don't think the question is specifically to do with living in a multi-racial country, more to do with the fact that social media amplifies the voices of complainants ten-fold.
The term 'twitter storm' didn't exist ten years ago - but Sachsgate is a prime example of why the BBC needs to be careful.