As IGC states, there was no London connection for the well known W Richards of Liverpool:
First name/s: William
Surname: Richards
First Address: 152 Dale Street
City/Town: Liverpool
County: Lancashire
Country: United Kingdom
Other Addresses:
144 Dale Street
Information
Trade: Gun maker
Dates: 1851-1864
Notes:
William Richards (I), full name possibly William Henry Richards, was a gun maker in London who died in 1820. He may have been the William Richards of 114 Strand, London, or the William Richards of Blackfriars, London, or neither of them. His son, William Henry (William (II)), moved in 1820 to Birmingham where he reportedly established his business in Primrose Hill. In 1842 he moved to establish a retail business at 152 Dale Street, Liverpool. The 1851 census records William (II) as a 55 year old gun maker (b.1796 in London), he was a widower living at 153 Dale Street, and from 1851 to 1858 he was recorded was recorded in business at 152 Dale Street, and from 1859 to 1864 when he died, at 144 Dale Street. William (II) had at least two sons, Henry Richards (b.1820 in London) and William Henry Richards (b.1828 in London). The 1851 census records Henry as a 29 year old gun maker living at 11 Aston Street, Birmingham, with his wife Hannah (b.1821 in Birmingham) and their children, Jane (b.1842), Susanna (b.1845) and Westley (b.1847). All the children were born in Birmingham which suggests that when William (II) went to Liverpool to open the retail side of the business, Henry remained in Birmingham to run the manufacturing operation. Also recorded living with Henry and Hannah was Henry's younger brother, William Henry, aged 23 and described as a gun maker.
It is reported that William (II) established a branch in Melbourne, Australia in 1853, and indeed there was a firm named William Richards which was recorded from 1856 to 1886 in Melbourne, and they claimed establishment there in 1853. It is almost certain that the firm was in fact established by William Henry who, as the younger son would not have inherited the Liverpool or Birmingham businesses. He has not been found in any subsequent census and may not have returned to England. William (II) was recorded in 1857 at 51 Oldhall Street.He appears to have retired or died in that year because Henry was recorded 51 Oldhall Street from 1858. An address of 53 Oldhall Street has been seen on a gun bearing the name W Richards so it would appear that he owned or rented both 51 and 53 Old hall Street. In the 1861 census Henry was recorded living at 61 Old Hall Street with Hannah and their children. Henry and Hannah were probably married in about 1840 because Jane had been born in 1842 and a previously unrecorded daughter, Emily, had been born in 1850. Susannah (b.1845) and Westley (b.1847) were not recorded and both appear to have died. Another son named William had been born in Birmingham in 1853. It would seem that William and Hannah moved to join William (II) in Liverpool between 1853 and 1855 because another daughter living with them in 1861 was Suzanna who was born in 1855 in Liverpool as were two more sons, Henry (b.1857) and Westley (b.1859). The Henry born in 1857 seems to have died because the 1871 census below shows a Henry (Harry) born in 1863/4. The name "Westley" is frequently found together with the name "Richards", but although a connection is likely with the famous Birmingham firm of Westley Richards none has been found. In the 1871 census Henry and Hannah were recorded living at Brickfield Cottage, Lower Lane, Fazackerley, Liverpool together with their children, William (aged 18), Susanna (aged 16), Westley (aged 10) and Henry (aged 7) were recorded but Emily was not.
In about 1878 Henry moved the firm to 27 Old Hall Street and in 1878 William (III) opened his own business at 44 Fishergate, Preston trading under the name of William Richards. One report states that the Preston business opened in 1872 but as William was only 19 years old at the time this is unlikely. The 1881 census records Henry and Hannah living in Lime Grove (no number stated), Litherland, Liverpool. Susanna, Westley and Harry were living with them. As in previous censuses, no occupations were recorded for the children. In the 1891 census Henry, Jane and Westley were recorded living at 2 Neville Road, Waterloo, just north of Liverpool. Hannah had obviously died between 1881 and 1891. Westley was recorded as a 32 year old gun maker. In about 1893 the firm opened a shooting ground in the Walton area of Liverpool. In 1895 Henry died, and Westley took over the Liverpool business. From about 1900 the business was recorded as William Richards.
The 1901 census records Westley living at 7 Morningside Road, Bootle, a couple of miles away from his previous address. He was married to Lilian (b.1871 in Huddersfield). In about 1905, one report states 1910, the firm bought the business of Williams & Powell at 27 South Castle Street, Liverpool. At about this time the firm appears to have had an address at Phoenix Chambers, as did Williams & Powell. From this time the firm claimed to have been established in 1780, but this was the date Williams & Powell were established rather than the firm of William Richards. In 1907 the firm moved from 27 Old Hall Street to Manchester Buildings, 1 Tithebarn Street, they also opened a workshop in Westmoreland Street. In the 1911 census Westley and Lilian were recorded living at Brooklands, Hayle Road, Waterton. By this time they had a daughter, Alice Lilian (b.1902 in Bootle) and a son, Arthur Westley (b.1906 in Waterloo). In 1913 the firm in Liverpool became a separate limited company, W Richards (Liverpool) Ltd. The Preston business remained trading as William Richards but in 1917 it became William Richards Ltd. In 1917 the company in Liverpool moved to Mellor's Buildings, 30 Exchange Street East (formerly Old Hall Street), and they took additional premises at 8b Rumford Place. At this time the firm had shooting grounds at Aughton, near Ormskirk. This shooting ground was roughly halfway between Liverpool and Preston.
In 1920 William died, and in 1924 the Preston business was moved to 6a Lune Street, it appears that by 1920 the separate company, William Richards Ltd, had been closed down and all the business in Liverpool and Preston was done in the name of W Richards (Liverpool) Ltd. In 1924 the firm bought the business of James (H?) Hooton of Liverpool. Westley retired in about 1935 (he died in 1944) and his nephew, Lawrence Richards Hunt took over. In about 1940 the company took additional premises at 22 Highfield Street, the purpose of these is unknown but may have had something to do with the bombing during the Second World War; these premises were closed in 1942. In 1943 Lawrence Richards Hunt retired and a J L Brown took control of the company.
In 1957 the company moved to 30 Moorfields, and in 1967 they moved to India Buildings, 42 Brunswick Street. In 1984 the firm became W Richards Ltd, so presumably the business was sold or the old company went into liquidation. The firm ceased trading on 30 June 1996, but the name, records, manufacturing rights and intellectual property of the company were bought in 1999 by Chris Caine. The company now trades as W Richards (Liverpool) Ltd at The Pavement, Pocklington, York, Yorkshire, YO42 2AX; Tel & Fax: 01759 305088.
Thereafter there are a number of potential W Richards listed as being in London and/or Birmingham. Given that you indicate only one apparent proof mark visible on the hammer gun, my assumption would be that comes from the 1813-1855 proof period. I have thus restricted my search to include only those gunmakers active at that time:
First name/s: Joseph & William
Surname: Richards
Location
First Address: 49 New Street
City/Town: Birmingham
Country: United Kingdom
Other Addresses:
112 New Street
16 Staining Lane (62 Strand), London
Information
Trade: Gun & Pistol Maker
Dates: 1799-1817
Notes:
Joseph and William Richards were recorded as gun and rifle makers at 49 New Street, Birmingham in 1799, but the business may have started earlier. In 1808 the firm opened an office at 16 Staining Lane, London. The partnership appears to have ended in about 1815 when William left, retired or died, and Joseph appears to have operated on his own in Birmingham until 1830. Whilst located at 16 Staining Lane the firm also apparently had premises at 62 Strand, some pistols bearing this address have been seen. It has been suggested that William Richards was the person who established the business of William Richards of Liverpool, in which case the firm of Joseph and / or William Richards may have been established in 1780.
First name/s: William P
Surname: Richards
Location
First Address: 20 Market Row, Great Portland Street
City/Town: London
Country: United Kingdom
Other Addresses:
13 John Street, Oxford Street
11 John Street, Oxford Street
Information
Trade: Gunmaker
Dates: 1823-1830
Notes:
Established in 1823 at 20 Market Row, Great Portland Street, he moved in 1825 to 13 John Street, Oxford Street. In 1830 he moved to 11 John Street, but appears to have ceased trading in that year.
Hope that helps narrow things down for you.
Tim