Born in Roxbury, Wheelwright was educated at Roxbury Latin and Harvard where he graduated in 1876. He studied architecture at M.I.T. and later in Europe, after which he worked in the offices of Peabody and Stearns and later McKim, Mead, and White.
Wheelwright and his partners at Wheelwright, Haven & Hoyt led the designs of a number of Boston area landmarks in a variety of styles, including the fanciful Harvard Lampoon Building, the Italianate fire station & tower in South Boston that is now the Pine Street Inn, and the Longfellow Bridge, where he helped perpetuate the story of Viking settlement along the Charles.
Serving as the Boston City Architect in the 1890s, Wheelwright saw the Boston Public Library completed during his tenure. He was selected by the Museum of Fine Arts to undertake a tour of European museums to inform the plans for the new building in the Fenway, and is thought to have contributed that learning to Guy Lowell's design.