Widening the picture a bit, the US, UK and Canada system was already in place during World War II. Intelligence was good about the Soviet Union by the end of the war. Along with knowing what the US had sent the Russians and the surrender of Reinhard Gehlen, head of Foreign Armies East, and part of his staff to the Americans. They would form the core of West German intelligence under CIA direction and control (Partners at the Creation by James Critchfield).
Let's now go over the problem areas. The Russians, Americans and British had German jet fighters, bombers, V-1s and V-2s plus nerve gas. Three types: Sarin, Soman and Tabun. I have seen two photos of German artillery shells filled with nerve gas recovered from boxcars by the Americans. The British found some as well and they were identified by an additional colored ring by British bomb disposal. They acquired an unknown number. Tests on live subjects began at Porton Down. Meanwhile, recovered documents, equipment and personnel were being gone over with a fine-toothed comb. The Russians had captured the almost complete DFS 346 supersonic aircraft, which they would test fly using an interned American bomber in much the same way the Germans had air-launched V-1s during the war.
Great Britain had no capability for testing long-range systems in the UK. That would be handed off to Canada and Australia. The Americans would offer limited assistance except in the area of atomic/nuclear weapons. In May 1946, one year after the end of the war in Europe, and supposedly, with no money, the US would start the Nuclear Energy for the Propulsion of Aircraft program. A report would also be issued the same month titled Guided Missiles and Pilotless Aircraft which stated, in part, that developments that would normally take years would now have to be completed in months. Why the rush? Development and testing of the X211 engine continued throughout the 1950s with successful operation.
Any good ideas the Germans had, along with setting up the White Sands site in the US, with German help, meant that 'other ideas' could be explored. The Russians were also doing all of the above. That is where the parity idea comes from. And though information is incomplete, Russian spies were well placed in the United States. They would at least notice unusual activity, the movement of heavy equipment, and new construction, especially at restricted sites.
If we look back at this from today, the Russians were playing catch up after the war ended. They had little excess capacity to develop advanced weapons during the war and knew they were behind the West in many areas. This resulted in their starting numerous programs to develop technology to rival the West ASAP.
Their starting point was a combination of technology given them by the West under Lend-Lease and stuff they captured from Germany at the end of the war. In both cases, this was somewhat haphazard in nature as they found they had lots of gaps in what they were starting with.
In the West, France and Britain resumed/ continued development of military technology. In France's case, often using German materials as a starting point, again to catch up. Britain was far more selective as the budget to do R&D was limited. In the US, wartime advanced projects were scaled back but continued almost unabated.
For the US, unlike the Russians, much of the necessary advanced technology was already in progress and captured German materials and research was more gravy than central to advancing projects.
As one example of this, all four countries mentioned had surface-to-air missiles in development in 1945. The state of these programs varied. The German technology in this area was actually limited, and most of it obsolescent.
Germany had a number of SAM missiles in testing by the end of the war. Most were subsonic making them obsolescent at best, obsolete at worst. Russia captured two: Schmetterling and Wasserfall. None of the German missiles had a guidance system that was operationally viable and such systems were proposals only by the end of the war.
The British had Brakemine (Army) and Stooge (Navy) under test. These were very basic subsonic missiles, but they had complete systems with guidance. Development would move to better designs resulting in Thunderbird, Bloodhound, and Sea Slug by the late 50's.
The French took captured German systems and started trying to improve on them. They abandoned these by the mid-50's for better designs that were mostly or entirely homegrown.
The Russians took the two missiles they had and started trying to make them viable. Schmetterling was dropped PDQ while Wasserfall became the focus on a first system. By the early 50's it was obvious that Wasserfall was a poor design and wouldn't work no matter how much it was modified. Focus shifted to what became the S-25 Berkut. The guidance system was largely a homegrown one that relied on British and US radar technology for the most part as the basis for that part of the system and really something of a tour-de-force of design that ended up grotesquely expensive to produce. A number of designs emerged from that beginning that were wholly Russian like the S-75.
In the US, there were three distinct programs run, respectively, by the Army, Navy, and Air Force. The Army projects started in 1944 and were Nike and Hermes. Hermes was an attempt to use Wasserfall and develop it by GE. The project as a SAM was short-lived and by 1947 had shifted to purely ballistic missile research, Wasserfall being dumped for the same reasons the Soviets got rid of it. Nike was the other program and it became the first operational SAM system in service anywhere.
The USAF did some ABM studies and quickly discovered that defense against a ballistic missile, even the V-2 wasn't possible in the late 40's and even into the 50's. Their other program, MX 606 GAPA was more an attempt to 'keep skin in the game' to get SAM's assigned to them in their break from the US Army.
The USN had two wartime systems that went to test designs that could be called at least marginally usable. These were Little Joe a quickly slapped together SAM that went no further than a few test firings, and Lark a first attempt at a really viable, if subsonic, SAM. It continued testing into the early 50's simply because it was available.
The USN project that went somewhere was Bumblebee. This was a truly massive and concerted effort to build an effective SAM. It started in 1944 and prior to the end of the war was firing test ramjet missiles at up to close to Mach 2 at a 12 to 15 mile range and 35,000 feet. Concurrent parts of the program developed new and far more effective solid fuel boosters, missile flight control systems that are still viable today, supersonic jet engine inlet designs, and fire control systems that could accurately guide a missile to an intercept at up to 75+ miles from launch. Out of Bumblebee came Talos, Terrier, and Tartar, and from them, Standard that is used today.
None of the US programs, outside the briefly lived Hermes, relied on German technology for much of anything.