Are UK speakers still using the auxiliary verb "shall"?

"Shall I bring you a cup of tea?"

I was learned to say this phrase in the approximately meaning of "Should I bring..."
But if the verb "shall" is not being used anymore, may be I can say: "Will I bring you a cup of tea?" Although it seems strange a little in that meaning, doesn't it?

In short I wonder if I can say "shall", "should" or "will" in the sentence above and what the difference is.
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Comments  (Page 4) 
Hello, Grammar Geek!

In spite of everything I appreciate your help very much. Thanks!

But could you explain this phrase to me, "To carry coals to Newcastle"? I don't quite understand what it means. Is it a proverb?
It's an idiom here. There IS already a lot of coal in Newcastle, so to carry coal to Newcastle is to provide something that it quite unnecessary. Like bringing fresh water to a well. Or bringing a cake to a pastry convention.
Students: We have free audio pronunciation exercises.
Grammar GeekI was just carrying coals to Newcastle at the point...

I wouldn't worry. Most threads develop a strong coals-to-Newcastle element, once they get to page 2...

Grammar Geek, thank you! I've got it.