Watson was born in Cumberland, being christened on 8 March 1804 at Hawksdale, near Dalston. His parents were prosperous farmers, who also owned an iron-forge. Although he had artistic ambitions from an early age, at his parents' insistence he was articled to a solicitor in Carlisle in 1821. Following his fathers's death in 1823, he abandoned the legal profession and went to London to study sculpture. He took advice from John Flaxman, and studied under Robert William Sevier and at the Royal Academy. In 1825 he left for Rome. On his return in 1828 he was determined to set up as a sculptor on his own account, rather than work in another artist's studio. However financial difficulties forced him to seek employment with Sir Francis Chantrey. He argued with Chantrey and afterwards worked for Richard Westmacott, William Behnes and Edward Hodges Baily. According to his biographer, Henry Lonsdale, he then spent two years at the Coade Artificial Stone Works in Lambeth, where he modelled sculptures and friezes for both private and public buildings. The work was well paid, but he decided to leave and set up his own studio once more.
He fell into financial difficulties again, and in 1832 had his belongings distrained for rent. However, eventually circumstances improved: a stone frieze for Moxhay's Commercial Hall in Threadneedle Street in the City of London, completed in 1842, brought him critical acclaim. Five and a half feet tall and seventy three feet long, it can now be seen in Napier Terrace, Islington. In the same year he received a lucrative commission from Lord Eldon for a marble double portrait of his grandfather, the first Lord Eldon, and his great-uncle, Baron Stowell. The commission had originally been given to Sir Francis Chantrey, who died before it could be carried out.
In 1839 Watson submitted designs to both competitions held by the Nelson Memorial Committee for a monument to be erected in Trafalgar Square. He was unsuccessful, but was later chosen to sculpt the relief panel of the Battle of Cape St. Vincent on the pedestal of the winning entry, William Railton's Nelson's Column. He was, however, unhappy with the terms of the commission, writing in a letter " I think the world will be disappointed with the relievi. The subjects are unfit for sculpture, or at least unfavourable." Watson died before the Cape St Vincent relief could be finished, having suffered from a persistent heart condition for most of his adult life. He also left the Eldon sculpture and a statue of John Flaxman to be completed by others. Just before his death he had most of the models in his studio destroyed.
- Double portrait of brothers Lord Eldon and Lord Stowell at University College, Oxford (1842)
- Statue of Queen Elizabeth I for the Royal Exchange, London (1844)
- Marble statues of Major Francis Aglionby (1843 - Aglionby was MP for East Cumberland) and Lord Lonsdale (1845), in Carlisle
- Statue of Flaxman for University College London (1843–1847)
- Pedimental sculpture for the Victoria Rooms in Bristol
- Frieze from Moxhall's Hall of Commerce in the City of London Installed in Napier Terrace Islington in 1975.
- Lonsdale 1886 p,89
- Lonsdale 1886, p.111
- Lonsdale 1886 p,143
- Lonsdale 1886, p.191
- Lonsdale, 1886 p.221
- Lonsdale, Henry (1866). The Life and Works of Musgrave Lewthwaite Watson, Sculptor. London. Retrieved 31 May 2011.