Hawker's P.1121 studies suggested that a Medway was most appropriate for low level strike fighters, the de Havilland Gyron most appropriate for a high-level interceptor, and the Olympus was best suited to a multirole fighter that might be called on to do both jobs.
The Medway was a low-bypass turbofan, similar to a larger Spey (or rather, the Spey was a scaled-down Medway), and wouldn't have worked well as a supersonic cruise engine on the Concorde. It might, however, have been a good replacement for the Conway on the VC10.
Pity.
Extract from Scenario 1957 of Project Cancelled by Derek Wood
Bill Gunston in his book Back To The Drawing Board - Aircraft That Flew But Never Took Off has a chapter about the Bristol 188 supersonic research aircraft. In it he says that Specification E.R. 134 which the aircraft was built was originally for a Mach 2 aircraft that could fly at that speed for at least 30 minutes, but it was altered later on to Mach 2.6 to support the Avro 730. The change in the specification meant that stainless steel had to be used instead of aluminium. Gunston wrote that the change probably doubled the time taken to build the aircraft and more than doubled the overall cost. Puddle welding had to be used and the aircraft was finally completed 5 years behind schedule. He also wrote that the Bristol Olympus could have been used, but the Ministry decided on the P.S. 50 version of the De Havilland Gyron. The first aircraft XF923 was rolled out on 24th April 1961, it began its taxiing tests in February 1962 and made its first flight on 14th April 1962. 3 aircraft (including the static test airframe) were built, but the Ministry wanted 6.Inevitably, the big high altitude bomber, the Avro 730 has to go, but to ensure long-term supersonic know-how, design and research contracts are issued to A.V. Roe for a Mach 2.5-Mach 3.0 experimental aircraft with sufficient fuel tankage for sustained supersonic flight. On this vehicle many of the problems of Concorde are destined to be sorted out.
Unfortunately the Bristol Type 188 got nowhere hear holding Mach 2.0 for half an hour and never reached Mach 2.6.
With hindsight the first 3 aircraft (including the static test airframe) should have been built of aluminium with the Olympus engine to the original specification for a Mach 2 aircraft. The second trio (which according to Wikipaedia were added to support the Avro 730 programme) should have been of a Mk II version, that is the aircraft actually built IOTL, but with Olympus engines. The Mk I could have been flying as early as the second quarter (1st April to 30th June) of 1957 and that might be early enough for any lessons learned to be read into the definitive Mk II. After the Avro 730 is cancelled it might be possible to modify them into test beds for the versions of Olympus used by TSR2 and Concorde.
Following Wood's Scenario 1957 and/or my suggestion won't save Concorde from being withdrawn when it did, but it might reduce the R&D cost and mean it entered commercial service sooner.
However, the reduction in the R&D cost might only be a in the bookkeeping. That is all the money saved on developing the Concorde would be transferred to the cost of the Type 188. Furthermore the Bristol 188 was an all-British project, while Concorde was an Anglo-French one. Therefore the French taxpayer would receive a disproportionate amount of the financial benefit.