Just a wee bit of supporting info about the fitting of British turrets to US aircraft. The fitting of British turrets to US aircraft for RAF service is not so well documented, but it seems that armament was fitted once the aircraft arrived in the UK, although I have seen a photo of a Ventura with the the Boulton Paul Type C Mk.II turret, the two-gun one, undergoing trials in California. Initially the Hudsons, the first of the flood of US aircraft for the British forces in 1939, were delivered by sea to Liverpool and erected at Speke. All of these aircraft had their serials assigned on the production line, so arrived in the UK with their identities firmly established in documentation.
From what I understand about the Type A turret, the BP four-gun turret of the type fitted to the Defiant, on foreign aircraft it was fitted in the UK. The turret saw use on Liberator Mk.IIs as the A MK.IV as well as Martin Baltimore Mk.IIIs (A Mk.VA) and Ventura Mk.IIs (A Mk.V). I'm presuming this work was done at MU level. Using the Liberator as an example, the Mk.II Liberator was delivered to the UK unarmed and had the BP turret fitted aft of the wing box, not in the common position behind the flight deck as on USAAF Liberators. The Liberator Mk.II is unique in this. Some were fitted with a BP type E tail turret, of the same type as fitted to the Halifax, the Mk.III variant fitted to the LIberator Mk.II, although not all were actually fitted with the tail turret, as I've seen photos of Lib Mk.IIs wth the Type A turret, but with no tail turret.
Boulton Paul seems to have been the biggest supplier of turrets to requirements for foreign aircraft operated by the British to begin with, as Nash and Thompson primarily supplied to the RAF's heavy bomber needs, with the exception of the Halifax. This is, from what I can gather because BP turrets were entirely self contained; they didn't rely on the aircraft's hydraulic or electrical systems to operate, which meant their incorporation into the aircraft was less intensive. First fitted in the Defiant, the Type A four gun turret was the first BP electro-hydraulic turret and was based on the SAMM De Boysson turret designed in France, only refined for mass production. The Type C turret was designed as a nose turret for the Halifax, with the C Mk.II being configured with a new cupola and for use as a mid-upper turret. Otherwise the mechanism was identical - all of BP's early turrets were the same as the first Type A in mechanism and design.
Determining serial numbers of exactly which Ventural Mk.IIs received the BP A Mk.V turret or the C Mk.IV turret going to be difficult unless there is a record somewhere at MU level of which aircraft was fitted with which turret. That'll require trawling through the ORBs and other such documentation at The National Archive, if such a record exists. Presumably the turrets were fitted owing to availability and corresponding to aircraft deliveries to the UK under Lend Lease.
The other route would be to go through Ventura Mk.II Aircraft Movement Cards at the RAF Museum to determine which MU the machines went to and to subsequently cross reference between ORB info and the aircraft's movement, which would be lengthy and probably costly as well. The difference between Mk.I and II Venturas was their engines, from what I can establish. Both were assigned simultaneously within the same first batch of 300 aircraft in mid 1940, with a second batch of Mk.IIs in late 1940.