A while ago I asked AQA how many bricks are in Stockport viaduct, they replied "The 27-arch 33m Stockport Viaduct was built in 1839 using 11 million bricks." At the time it carried two tracks and was later doubled in width. Harringworth Viaduct has 82 arches and a maximum height of 18m, and also carries two tracks, so if AQA were approximately correct, 20 million bricks sounds at least of the right order of magnitude.
You could come up with a lower bound by working out the surface area of all the masonry one one arch and dividing by the area of the external face of a brick, then multiplying by 82 for the arches.