(Up until then, "memory" was anything but dynamic. Programs were typically hard-wired into the machine. The concept of having a general-purpose computer, which could read in a program from an external source was laughable science-fiction.)
It stopped being laughable science-fiction beforn Kilburn's work. The credit for the theoretical work/design should go to John von Neumann (w/Goldstine, Mauchly, & Eckert to some degree) for his "First Draft of a Report on the EDVAC", published June 30, 1945, and written a few months before that. That document lays out exactly how to build the aforementioned general-purpose programmable computer. The EDVAC was in fact never completed, but nearly every computing group after 1945 followed the First Draft's design.
It's certainly true that Kilburn et al's work was particularly important. His group (directed by M.H.A. Newman, the man who taught Alan Turing Godel's Incompleteness Theorem) was first to make the EDVAC report an engineering reality, beating the Americans by getting a prototype up and running in June 1948.
What's interesting is that the Williams tube's days were seriously numbered, when Jay Forrester of MIT published "Data Storage in Three Dimensions" in April 1947, presaging the use of the magnetic core memory which supplanted the CRT tube. MIT's Whirlwind and the Manchester Mark I were basically built in parallel, along with the other computers of the time, such as the JOHNNIAC and IBM 700.
The ENIAC was (more or less, excluding Konrad Zuse's work, e.g.) the first general-purpose computer, but required hard-wiring instructions. It was designed in 1943/44 and ran its first problem in summer 1945; it was declared complete w/documentation on June 30, 1946. It proved that a computer using thousands of vacuum tubes would work, whereas many had predicted that the failure rate would be too high.
Sources for the above include The Computer from Pascal to von Neumann, Goldstine, and Alan Turing: The Enigma, Hodges.