we need to solve it! let's put our thinking caps on!
is that idiom used in the British English?

Question

we need to solve it!  let's put our thinking caps on!
is that idiom used in the British English?

Answer

Yes, you can say "put your thinking cap(s) on".

I'm not sure if it's a slightly old idiom but we certainly recognise and understand it.

Answer

And what about "To turn on the thinking machine", not referring on a super PC, but meant ironically?

Answer

I wouldn't say that it's old - fashioned. I hear people talk about putting on different caps quite frequently.

I own deBono's course called 'Super Mind Pack." I've been 'thinking' about using it for some time.

Answer

It is quite an old-fashioned one. I have not really heard people use it in a long time.

(Certainly don't think DeBono coined it; although he did talk about thinking hats)

Answer

Agreed. The idiom is known and occasionally used,  but it's quite old-fashioned.