This is what was published in Aug 2024 in a UK Govt paper about the Common Missile Compartment for the Dreadnought and Columbia classes.
"A Common Missile Compartment (CMC) for the SSBN, which will house the
current Trident strategic weapons system, is being developed in conjunction
with the United States.5"
"5 The US is replacing its current SSBN fleet with a new Columbia class SSBN, which is due to enter service within a similar timeframe to the UK’s Dreadnought programme (early 2030s). Design for the Dreadnought’s common missile compartment (CMC) is being delivered under the 1963 Polaris Sales
Agreement (PSA), as amended (HL Deb 11 February 2013, c92WA). Each SSBN will have three missile compartments, each with four missile tubes, for a potential total of 12 missiles per submarine"
And
"As outlined above, the Common Missile Compartment for the submarine is
also a collaborative programme with the United States. American company
General Dynamics is the prime contractor for the CMC and is working in co-
operation with BAE Systems to ensure that the
design accommodates UK
requirements for the Dreadnought class. In October 2016 Babcock
International was awarded a contract by General Dynamics to manufacture
22 tactical missile tubes as part of the overall CMC project. That work is
taking place in Rosyth, securing approximately 150 jobs. Whether this work on
the CMC forms part of BAE Systems’ UK supply chain spend, is unclear."
And
"As with previous SSBN,
the submarine will be built by BAE Systems in Barrow-
in-Furness and the propulsion system will be built by Rolls Royce at
Raynesway, Derby."
(my emphasis in bold)
Note the CMC for both classes comes in 4 missile blocks - 3 in Dreadnought & 4 in Columbia.
This article refers to various contracts to GD for development and "installation" works on various systems in both classes. Doesn't say where the installation will occur.
The Dreadnought-class ballistic missile submarines will help maintain the UK’s Continuous at Sea Deterrence (CASD) capability.
www.naval-technology.com
So while development is by General Dynamics in the US, actual construction of the missile blocks is less clear, but some elements of those blocks are being produced in the UK. But I note the reference to "the submarines" being built in the UK, not that only part (I.e. entire sub less missile compartment as suggested by Z42) will be.
That is the problem nowadays. Where do all the bits come from? So how much is "construction" and how much is "assembly" of components / blocks "constructed elsewhere.
Specialist steel for the Dreadnoughts is being sourced from France.