It was up to individual owners to fit their ships with weapons – that would inevitably mean what they could afford.
Lloyd’s Register of Shipping shows the armaments. For this period they may be found [
you will have to scroll down the page] here:
https://hec.lrfoundation.org.uk/archive-library/lloyds-register-of-ships-online– see as example =
https://archive.org/details/HECROS1780/page/n7/mode/2upColumn 7 shows the name of the owners, and some entries also shows the armament
According to David Syrett [
Shipping and the American War], which studies the hiring of ships to transport stores or troops to North America, they initially had difficulty in getting shipowners to arm their ships at all. This was because it would contravene an Order which prohibited the export of guns and ordnance stores to America. Syrett states that most ships going to America 1775 – 76 were unarmed. After 1776 an Order in Council lifted the restriction.
After this, for merchant ships hired for service as transports or to carry stores for the Government, then the Admiralty did lay down a minimum – owners should arm ships with “at least six carriage guns of six pounders or less bore, according to the size of the ship” and later [1779] allowed shipowners to fit carronades ‘not less than 12 pounders’
Looking at the details of those vessels either taken into Naval service, or hired as a cargo/troop transport, then it would seem that the usual calibre was 4- or 6- pounders, although some were able to carry 9-pounders. The number carried seems to have increased, but would obviously depend on the size of the ship.
For examples, see:
https://discovery.nationalarchives.gov.uk/details/r/C10608746https://discovery.nationalarchives.gov.uk/details/r/C10609363