Well I'm going to give you my gut opinion on these aircraft, all what-ifs have an element of unknown about them, especially paper designs that never flew at all.
BAC TSR.2: (I won't call it the Eagle because there is no proof of that being an official name, TSR.2 was cancelled before the naming panel would have even sat down to ponder choices). I believe that had all the potential to be a great bomber, BUT the electronics had the potential to be a nightmare too, its hard to image the RAF getting any until around 1967-68, possibly ironing out the snags into the early 1970s and then probably stripping out the analogue systems in the late 1970s at even more expense. Tornado was long in development but when it arrived it was the latest standard, by 1970 TSR.2 would have still had 1960-era analogue systems with digital converters etc. Exports unlikely anywhere except Australia (if political intrigue from UK and USA hadn't killed it), other export unlikely due to security issues, nuclear capability, cost and US interference. Had Tornado not existed what Germany and Italy would have brought as a strike aircraft and what Britain would buy as a long-range fighter is up for grabs. Cue American imports or some kind of Franco-German-Italian design.
Westland Westminster: Westland seemed sure about low running costs for the civil Westland with the Napier Eland. The basic Sikorsky S-64 transmission was probably an Achilles heel too. The main problem was lack of interest, the helicopter had no real interest from BEA at this time, despite earlier plans, and the RAF weren't so bothered either and what official interest there was, ended up direced at Rotodyne. Westland designed other helicopters, some of them tandem-rotors like the Chinook but in the late 1960s the MoD ordered Chinooks, only to cancel the order due to budget cutbacks (also paying cancellation payments) and then re-ordering them again in 1978 at increased cost. What killed the Westminster was the lack of interest, had Rotodyne not existed then maybe it would have been brought. Even so there was a twin-rotor bias at this time for heavy lift, but it certainly could have avoided buying Puma in the early 1970s.
P.1154/P.1216: P.1154 was pie in the sky. It was impossible for the RAF and RN to combine two different fighter needs into one common airframe. At the time buying Phantom for the RN made the most sense. At that time the Invincible Class was merely an escort cruiser design, CVA-01 was the carrier of choice, a proper carrier capable of handling proper jet fighters. For the RAF the Harrier was small and easy to operate in the field. P.1154 with afterburning nozzles, meant hot-air ingestion and serious ground erosion (not to mention melted tarmac and similar problems on carrier decks) which would have been serious problems for VTO. STOL was perhaps more likely but Jaguar was probably far superior in actual load-lugging capability. Derek Wood felt the earlier P.1150 might have been better as it shared more structure and the Pegasus engine of the P.1127 and had a sufficient Mach 1.2/1.3 dash speed. Exports unlikely, Harrier didn't sell that well and I've never seen mention of any US interest in P.1154. P.1216 was the culmination of a whole series of similar designs during the late 1970s and early 1980s. It built on technology from the P.1154, but it needed a new engine (the RB.422), carbon-fibre fuselage with lithium-based aluminium alloys and superplastic formed/diffusion bonded titanium parts (as used on Eurofighter). It could have been as good as any 1980s fighter but lets be honest a very expensive project that might have not entered service until the late 1990s and one that needed UK/US collaboration, which wasn't forthcoming. It was not a serious rival to the European EFA or the US thinking that eventually led to JSF. I still feel supersonic VSTOL aircraft are a pointless exercise, a hang-over from the heady days of the 1950s. The F-35 doesn't really need a VSTOL variant, it adds nothing but cost and complexity.
Fairey Delta 2 and 3: Delta 2 was good, but it was a research aircraft built around the Avon jet engine. It had no fighter potential without totally redesigning it. So a British Mirage was not as simple as some would suggest. Also again there was no official interest in a Mirage-type fighter within the RAF. The Delta 3 was a beast, two Gyron jets and two Sceptre rockets, this beast was a seriously powerful aircraft, designed for Mach 2-2.5, only its structure was limiting the top speed, it could have probably reached Mach 3 with the proper materials. But, the radar and missiles were not great, Red Dean was a radar-guided all-aspect weapon but massive. British missile technology was not great and whether this missile would have been any good is debateable. With Firestreak the Delta 3 was nothing more than a more powerful two-seat Lightning. It would have been obsolete before it even flew, certainly no match for later 1970s aircraft and would have been retired in the 1980s like the Lightning. Phantom would have been better all-round and had Sandys not scrapped it in 1957 it wouldn't have got much further beyond 1960.
English Electric Lighting: Has its faults but was awesome, as a point-defence interceptor it was perfect. As the Soviets turned to stand-off missiles it was less useful. The planned automatic interception system was never declared operational, radar wasn't great, Red Top was ok but Firestreak was superior to Sidewinder in being able to engage supersonic targets from all-aspects. Never had the radar to support the planned radar-guided Firestreak variants.
Fairey Rotodyne: Noise was a problem but the Battersea trials in 1960 didn't reveal much of a problem. The Eland wasn't great but the bigger production variant would have had the Rolls-Royce Tyne and separate gas-generator turbines. The tip-jets were only used for ascent and descent, no real fuel problem. The problem was the Rotodyne was a flying proof-of-concept, yes it worked but it took time and money to get right and the merger of Fairey and Bristol helicopters and Saro into Westland caused delays and internal rivalries. The airlines were not serious, the RAF lukewarm and the MoS not too eager either. Export deals were agreed but for the full-scale version. The US customers, used to the US industry, expected it to appear within a year, it still hadn't within two so they pulled out. The production Rotodyne Z might not have flown until 1963. Civil exports were unlikely, city-to-city helicopter services never really took off anywhere despite the heady 1950s optimism. Militaries might have brought it, but it was a large and expensive bit of kit at a time when Vertol were rolling off hundreds of Chinooks for Vietnam.
Short Belfast: I agree should have been more built. But in reality the aircraft should have been the Handley Page HP.111C based on the Victor bomber. That was the design chosen by the RAF but the government, concerned with unemployment in Northern Ireland continued to prop-up Shorts and gave them the job. The Belfast was ok but less than ideal. Saying that the HP.111C would have only been ordered in small numbers and probably retired in 1975 to save money too.
Avro 770 series: The RAF wanted to replace the Shackleton, but not did not want to buy the Breguet Atlantic. They wanted something better, then wanted something equal to the Atlantic and then went high-end again. Nimrod was ideal for the GIUK gap role, good speed and loiter and even a strike role. Better than the P-3, the MRA.4 should have been better and we could have had it now rather than relying on a couple of C-130s flying about hoping to spot subs by eye. Might as well buy some bloody Catalinas!!
Vickers VC-10: There should have been earlier air-air VC-10 tankers and an AEW based on it. Skybolt missile carriers were going no-where past 1962.
Tornado: The ADV is NOT a failure as an interceptor! It did what it was designed to do, destroy long-range Soviet bomber way out to sea before they can launch their stand-off missiles. It didn't need to dogfight, it was designed to loiter and engage at long-range. Let the Jaguar and F-16s in Europe do all the dogfighting. Had Tornado not existed then Britain would have brought F-15s at seriously high prices, but Britain (and Europe) did not want that. They feared the US would destroy the European aircraft industry if Britain got a licence-deal to assemble F-15s. The cost was high too. F-14s even higher. Perhaps an improved Anglo-French Mirage 2000/ 4000 could have been developed though. The ADV was the perfect tool and Britain could actually afford it too.
British Phantoms: Agreed. Stupid decision to safeguard British jobs.
Westland Sea King: No, Westland made the shrewdest move of all the British firms in getting that Sikorsky licence-deal. Sea King was perfect and earned lots of export £s too.
Vulcan B.3: Could have been done, probably as a rebuild rather than new airframes. After Skybolt there was not much choice of long-range weapons. Polaris was the best choice. Blue Steel was never what it should have been and plans for longer-range and low-altitude versions never got very far. It's hard to see much future for the Vulcan and its amazing really they lasted until the 1980s. Long-range but by then past their prime with 1950s tech. An awesome plane but even the most greatest of aircraft has to exit the stage at some point.