But how can we do this politely and in a British way?

First, we can begin 'would like to' to say what we want.

'Hi i would like to speak to the manager. Yes, I'd like to complain.'

We can say 'I am unhappy about' plus a noun phrase.

'Yes, I am unhappy about my late delivery.'

We can also add 'very' or 'really' to make this stronger.

'Yeah, I'm really unhappy about this.'

We can also say 'it's unacceptable that' plus a clause.

'Yeah, it's unacceptable that my delivery is late.'

Remember, don't lose your temper.

This means 'don't become too angry' - it's rude to lose your temper.

This is good advice - we can say this to people who are angry with us:

'Alright! Don't lose your temper.'