Just to round up info on this gunmaker, here is what IGC has on them:

Name Alexander Thomson
Other Names Alexander Thomson & Son; Alexander Thomson; Alexander Thomson & Son
Address1 3 Drummond Street
Address2 16 Union Place
Address3 16 Union Place & 3 South St Andrew Street
Address4 16 Union Place
Address5 95 Princes Street
Address6 3 Hanover Street
City/Town Edinburgh
State/Region/Province Scotland
Country United Kingdom
Trade Gun maker
Dates 1820-1881

Notes

A family by the name of Thomson were armourers in Edinburgh from about 1600 to 1700, but they may not have been connected with Alexander Thomson. Alexander (R?) Thomson established his business in 1820 at 3 Drummond Street, Edinburgh, he was appointed "Gunmaker to His Majesty" (King George IV) in 1828.
In 1833 he took over the business of George Thomson, see below, (father or brother?) who was appointed "Gunmaker in Ordinary in Edinburgh" to His Majesty in 1828; he moved to George Thomson's address at 16 Union Place. In 1840 and 1841 additional premises were taken at 3 South St Andrew Street. In 1869 the name of the firm changed to Alexander Thomson & Son and they moved to 95 Princes Street. In 1876 they moved to 3 Hanover Street. They closed in about 1881. No trace of either Alexander or George Thompson has been found in the censuses.

Name George Thomson
Address1 4 Arthur Place
Address2 16 Union Place
City/Town Edinburgh
State/Region/Province Scotland
Country United Kingdom
Trade Gun lock maker; gun maker
Dates 1819-1832

Other Thomsons in Edinburgh:

Name Thomas Thomson
Address1 South Craig's Close
City/Town Edinburgh
State/Region/Province Scotland
Country United Kingdom
Trade Gunmaker
Dates 1804

Name John Thomson
Address1 14 Princes Street
Address2 3 South St Andrew Street
City/Town Edinburgh
County Midlothian
State/Region/Province Scotland
Country United Kingdom
Trade Gun maker
Dates 1820-1827
Notes
John Thomson was probably the son of Thomas Thompson of South Craig's Close, Edinburgh (1804), and brother of James Thompson; both were apprenticed to James Innes, James in 1793, John's date of apprenticeship is not known. In 1820 John bought the firm of James Innes & Co at 14 Princes Street, Edinburgh, and commenced trading under his own name. In 1824 he moved the business to 3 South St Andrew Street, but he closed in 1827.

Name James Thomson
Address1 James' Court
Address2 Skinners Close
Address3 Cannongate Head
Address4 Parliament Square
Address5 9 Parliament Square
City/Town Edinburgh
County Midlothian
State/Region/Province Scotland
Country United Kingdom
Trade Gun maker
Dates 1804-1815
Notes
James Thomson was probably the son of Thomas Thomson of South Craig's Close, Edinburgh (1804), and brother of James Thomson; both were apprenticed to James Innes, James in 1793.
He was recorded as a gunsmith in James' Court in 1805, and in Skinners Close in 1806. In 1807 his address was Cannongate Head, and in 1809 it was Parliament Square. From 1812-1815 his address was given as 9 Parliament Square, but he was not recorded after that date. On 9 March 1814 he registered patent No. 3784 for a single barrelled multi-chambered weapon.

Tim