In the US there is a slight difference in meaning. “Have” for something closer and better in a question”, “take” for something further away and better for the imperative. People are crowded around a photograph and I can’t see it. “Hey guys, can I have a look?” Today’s crossword is a killer. Here, have a look and see if you can get any I’m stuck on. (Gives the other person the puzzle) (“take” would be fine too but I would slightly prefer have) I heard you got a new car. Can I see it? It’s in the garage. Go take a look. (“Have a look” in this situation would sound British to me)

I don't think that there's much of a difference in terms of politeness. And if there is, I'd say that it's actually the other way round -- to me, 'take a look' sounds marginally more polite than 'have a look'. The main difference is regional. American English tends to prefer phrases with 'take a x', while British English favours 'have a' or 'go for a' e.g. 'take a bath' (mainly AmE) v 'have a bath (mainly BrE) or 'take a walk' (mainly AmE) v 'go for a walk (mainly BrE).

Many thanks.

I’m not a native speaker but I too think they both have a slight nuance in meaning.

I'm in the UK and I wouldn't use the term 'take a look'. It sounds more American English to me. 'Would you have a look at this please?' would be polite/formal. 'Have a look' is more familiar, e.g. between friends.