My thoughts on the air side;
As will repeat what I said regarding the Ark Royal and Eagle, a 1957 decision comes at the wrong time, vast amounts of money are already tied up in recent refits of Ark and Eagle and the rebuilding of Victorious and of course the development and testing of the aircraft aboard them. How much will be saved in the end is open to question and whether the effects would in effect worse than the real Sandys Paper is a worthy question too.
Sea Vixen, the third navalised prototype flew June 55 and not due to enter service until 59, but hopefully earlier. By the time its ready the carriers won't need them so its a scrapped. The RAF won't want them given they have Javelin (which has its own flaws). Had the Admiralty not faffed about so much between 1950-55 then DH.110 might have been ready by 55 in which case the problem doesn't arise. The thin-wing work is wasted though.
Scimitar, only just flown 19th Jan 56, the first service example in Jan 57. What to do? Most likely outcome is cancellation as the FAA jet squadrons are wound up. Its possible the RAF continues to buy the type as a shore-based anti-ship bomber or perhaps as a dedicated ground-attack fighter and giving the RAF a tactical nuclear bomber for service in West Germany by 1960 and of course extending production to beyond the 76 built. Perhaps to the full 100 originally intendd and perhaps beyond. Supersonic developements might be followed but seems unlikley.
Hunter, the supersonic P.1083 should be completed to complement the Scimitar fighter-bomber force in Western Europe. Likely exports to Belgium and Netherlands in 1960-62.
F.155T, does this go ahead? Thin-Wing Javelin cancelled as not really going anywhere quickly and the basic design has shortcomings. The Navy still has SAM capability for self-defence but no carrier force to augment Fighter Command and of course AAM-armed fighters still needed for FEAF and MEAF. F.155T is too advanced for those areas and too expensive for mass production. I can't see the removal of the carriers to seriously impair defence of Britain as the RN is protecting itself more than the nation so the manned aircraft versus missle debate goes on. One side-effect of this scenario is that the Fleet is now protected by missiles only, so logically Sandys can argue a stronger case for SAMs over manned fighters. So F.155T seems doomed still. Out of the carnage perhaps P.1121 can be saved as a timely replacement for the Scimitar in the tactical nuclear role.
F.177D, the Saro P.177 is nearing first flight as the axe falls on the carrier fleet. Out goes the P.177 immediately. 150 P.177N chopped leaving just 150 P.177R for the RAF. If P.1083 has been followed the RAF might see the need to replace the Hunters as less important and Lightning coming on stream. Could the MoS justfiy the P.177 programme on the basis of 150 aircraft? Would W.Germany and Japan see the writing on the wall sooner and look at Starfighter earlier? I see P.177 being cancelled too, any need for a hybrid rocket-jet fighter seems much less important than an all-round supersonic fighter-bomber for FEAF and MEAF. Again P.1121 family might be salvaged.
Future Fighters, I see P.1121 as entering service in two forms, a nuclear bomber for RAFG and basic Red Top fighter-bomber for FEAF and MEAF and to replace Javelin serving alongside Lightning. The shift to tactical roles and the need for a long-range interceptor will probably lead to AW.406 becoming a pure land-based VG interceptor. The P.1127/1154 saga is simpler given the removal of a naval need unless the lack of a protective fighter for the ASW carriers is ever raised again in the mid-60s. Phantom seems unlikely given the lack of a naval need, BAC might proceed with a Type 583-esque fighter for entry into service in 1967. RAF likely to get cheaper Harrier too for RAFG with P.1121 soldiering on with Lightning until a 70s fighter is ready, perhaps Jaguar, perhaps something slightly like a Super-Jaguar. An improved Harrier like HS.1184 might be likely in the 70s if funds allow but TSR.2 will probably eat those funds during the early 70s. EFA then replaces all come 1990s.
Nimrod, would replace the Shacks, perhaps more of them to compensate for the loss of the carriers and to augment the ASW carrier fleet. A Comet or Nimrod based AEW seems more likley slightly earlier, perhaps operational in the early 70s.
Gannet, would continue in order to equip the Hermes and Bulwark until the mid-late 60s. There would still be a shore-based role for them across the world.
Buccaneer, due to fly in 1958, does the lack of carriers kill this before first flight? Probably I see no real RAF desire to take on the plane as it doesn't really fit as a Canberra replacement and thoughts towards TSR.2 are looking a generation ahead. Probably the end of Blackburn.
GOR.339, likely as not B.103A, DH.110 derivative, P.1121, Type 565 in the early stages of the GOR in an attempt to save these products. B.108 likley to follow and is likely to be a fresher design unconstrained by current production practice laid down by Bucc production and perhaps a freer hand to redesign. P.17 probably still the best contender. I'll leave the question of the EE Vickers merger alone for another what-if another day but TSR.2 seems certain to go ahead without any meddling from Mountbatten etc. I'll also leave aside any financial and political considerations of TSR.2 alone but certainly the lack of the carrier force makes it more certain TSR.2 will be brought for a global power-projection role although by 1970 its doubtful wheter there would be a FEAF and so numbers probably would be around the 90-100 mark. Of course there is no OR.346 afterwards.
Blue Envoy still seems likely for cancellation given the RAF's misgiving of its low-level aircraft performance. Another couple of batteries of Bloodhound seem likely.
Blue Water, might be more likely if no Scimitar or Bucc, less likely if the RAF used either of those as a tactical nuclear bomber. That would open the door for PT.428 which might still happen if the funding is freed for it by less Naval expenditure, itself a what-if given the ASW needs. An air-launched long-range ASM based on Blue Water sounds crazy but its an odd little what-if I'll throw in here.
Green Cheese looks likely to be cancelled, it can't fit Gannet and that platform is not a serious bomber and there is no RAF to operate it. Red Beard seems more likely, perhaps Tychon in the 60s until Martel ready.