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I would like to write the following in a British English style:

I’ve been told I’m not the greatest singer so if I were to say, “I’m going to be a famous singer,” you could reply, “Not a chance!”

Is my punctuation correct regarding the comma after singer? I know that I’m supposed to use a comma after the ‘if’ clause but I’m not sure if, in this case, it should be inside the quotation.

British practice is to place some kind of punctuation at the end of direct speech, before the single (usual) or double (allowed) quote mark at the end of the speech.

There should be a comma, full stop, question mark, or exclamation mark
at the end of a piece of speech. This is placed inside the closing
inverted comma or commas.

‘Can I come in?’ he asked.

‘Just a moment!’ she shouted.

‘You’re right,’ he said.

'I didn't expect to win.'

Punctuation in direct speech (Oxford)

If a comma isn't part of the quotation itself, and the narrative interrupts the quotation, then the comma should be outside of the closing quotation mark.

On page 16, the University of Oxford Style Guide says:

Place any punctuation which does not belong to the quote outside the quotation marks (except closing punctuation if the end of the quote is also the end of the sentence).

It gives some examples:

✔ After all, tomorrow is another day. 🠆

     'After all,' said Scarlett, 'tomorrow is another day.' OR

     'After all, tomorrow', said Scarlett, 'is another day.'

✘ 'The kitchen,' he said, 'is the heart of the home'.

✔ 'The kitchen', he said, 'is the heart of the home.'

In the first example, the commas inside the quotation marks are correct because the unquoted sentence has commas in those positions. But since the original sentence has no comma after tomorrow, the comma comes after the closing quotation mark in its second variation.

A further example:

✔ Bob likes cheese 🠆

     'Bob', I said, 'likes cheese.' OR

     'Bob likes cheese,' I said.

In the second variation, the comma comes at the end of the quoted sentence.

Given all of this, your example sentence should be punctuated in this way:

I’ve been told I’m not the greatest singer so if I were to say, 'I’m going to be a famous singer,' you could reply, 'Not a chance!'

The comma after singer is not part of the quoted sentence—but it comes at the end of the quoted sentence, so it is inside the closing quotation mark.

Note: My interpretation of 'Bob likes cheese,' I said. being correct (where it's the end of the quoted sentence, so the comma goes inside), is only right if it's not actually a typo in the Oxford style guide. Based on all of the other rules, I would have assumed that the end of the sentence is I said. (as opposed to likes cheese.'). So, in order for it to not be a typo, I'm interpreting the guidance as talking about the end of the quoted sentence in this case. I do have an open query with the Oxford style guide on this. Barring feedback, I can only assume that there is no typo. This is supported by the Oxford Dictionaries article referenced in the other answer.