Is the Brittish term "Bollocks" likened to the American term "Bullsh*t"?

I can but agree with yojimbo. “Ask me bollocks” is inspired. And also the use of the collective noun as a singular, as in “he’s a bollix” was also new to me.

‘Balls!’ said the Queen. ‘If I had to, I’d be king!’
The king laughed. He had to.
The prince didn’t. He was shy to.
The princess did. She wanted to.
The duke said they’re both nuts.

My Dictionary of American Slang has “balled up” as a synonym for “bollixed up” (confused, riddled with blunders), but no ballocks.

A random thought…is “nuts” (like Gen. McAuliffe’s response to the German surrender demand) part of this testicles/bullshit family?

Indeed it is.

I suspect the General was really thinking “Bollocks to 'em, Nazi bastards!”

According to Bill Bryson, the original version of the song “Barnacle Bill” was called “Ballocky Bill”, and the rest of the lyrics were far less innocent.

Don’t know how true that is, given some of the stuff I’ve read about him in GD.