This doesn't answer your question but, I have read that the RAF studied fitting its C-130Ks with Tyne engines, but decided that the improvement in performance was not worth the improvement in cost.
Also Bristol/BAC proposed a licence built STOL Hercules with Tynes to OR 351 called the BAC222, but was beaten by the AW/HS 681. However, if the RAF had selected the BAC222 and the licencing agreement had included exclusive sales rights in the Commonwealth and Europe (like Westland's licences on Sikorsky helicopters did) then there are some export sales too.
However, my preferred solution is this. I don't see why the British aircraft industry with all the British turboprops that were under development in the 1950s could not have developed a superior aircraft to the C-130A and put it into service with the RAF instead of the Beverley.
As the Beverley was a Blackburn product and it became part of Hawker Siddeley, I think Armstrong Whitworth should have developed it. 50 Mk 1 aircraft are built for the RAF instead of the Beverley. Then 56 Mk 2 plus 17 civil sales instead of the Argossy. Next 93 Mk 3 in place of the 6 HS 681 ordered, 66 C-130K and 31 Andover Mk 1. That's 216 so far.
Then the RAAF buys 12 instead of the 12 C-130A it bought in the 1950s and 12 instead of the 12 C-130E it bough in the 1960s. The RCAF buys it in place of the C-130B, E and H it bought in the real world and the RNZAF 5 instead of the C-130H it bought in the 1960s. The SAAF buys 7 in place of the C-130B it bought and possibly 9 in place of the Transals, depending upon when the arms embargo comes in and how strict it is.