Did Paul Revere shout "the British are coming"?
Almost certainly not, and the phrase would have made little sense in April 1775.
Why the wording is wrong. The colonists at that point still largely considered themselves British subjects. Announcing that "the British" were coming would have been confusing. The contemporary term for the troops was the regulars — professional soldiers of the regular army — and the warning is generally reported as something closer to "the regulars are coming out".
Why he would not have been shouting at all. The ride was a covert operation through countryside patrolled by British officers specifically watching for messengers. Revere's task was to reach particular people — notably Samuel Adams and John Hancock in Lexington — and to alert militia leaders quietly, house by house. Shouting through villages would have got him captured faster, which in fact eventually happened: he was detained by a British patrol before reaching Concord.
What the ride actually involved. Revere was one of several riders, not a lone hero. William Dawes rode a different route, and Samuel Prescott, who joined them later, was the only one of the three to reach Concord. The alarm spread through an organised network of riders and signals, not one man's shouting.
Revere's more significant contribution was arranging the signal from the Old North Church — one lantern if the troops moved by land, two if by sea — and being a central organiser in Boston's intelligence network.
Where the legend came from. Henry Wadsworth Longfellow's poem "Paul Revere's Ride" (1860) made him a national figure, eighty-five years after the event. Longfellow took considerable liberties — inventing details, omitting Dawes and Prescott entirely, and having Revere complete a journey he did not finish.
The poem was written on the eve of the American Civil War, and is generally read as a deliberate call to patriotic unity.