"Different but can be made to fit" would be my assessment from my little experience in this area. The butt plate profile is different, otherwise I don't think there's much if anything in it. SMLE ones were usually fitted with an armoury disc I think, certainly older ones. Other than that I would say that magazines are almost interchangeable although different. Foresight blades?
It's worth noting that the Savage and Long Branch No.4s at any rate have different screw threads, at least in places than the UK built ones. Foresight protector screw springs to mind, different length too, there are lots of detail differences.
Up to WW2 No1 rifles used Enfield dimensions for threads on screws, etc. The No was designed during the 1930s with simplified BA threads on most parts. Dimensionally, most of the key tolerances remained the same between No1 and No4 (bolt body, receiver group internals, etc). All of the five No4 factories (Fazakerley, Maltby, BSA, Savage. Long Branch) produced dimensionally identical parts - so everything interchanges.
In 1942-ish it appears that emergency instructions were issued to allow No4 parts - which were in mass production by then - to be used in the repair and maintenance of No1 rifles, which were no longer in production. Hence "Dispersal" No1s, and WW2-refurbished No1s, can contain the following No4 parts:
Some No4s, particularly the various Trials models, use converted No1 magazines.
In 1953-ish, BSA carried out a series of major No1 refurbishments. It seems BSA bought up thousands of obsolete No1 rifles and refurbed them for commercial sale. They also seem to have done this on UK government contracts for No1s to be supplied to various colonies and Dominions - e.g. South Africa. At this time, BSA manufactured a great deal of new No1 woodwork in beech. The butts they made were designed to be fitted to either No1 or No4 rifles - they were of a unique pattern that had the narrow "tab" cut for the SMLE buttplate, but lacked the cut-out for the safety catch. I.e. they were No4 at the front and No1 at the arrse.
The No1 and No4 barrels have identical screw threads, but they are indexed 180o apart - presumably to stop armourers trying to experiment with hybrid rifles!