My understanding is that the Tizard mission played no significant role in the Manhattan Project aside from setting up regular "channels" for communication between the National Defense Research Committee and the British scientists. At the time of the mission (September-November 1940) the British had not yet fully completed their MAUD Committee work (evaluating the feasibility of a nuclear weapon), and played it all rather close to their chest. They contacted a few scientists who were involved on the American side but did not tell them much they did not already know, and nobody changed their mind on the matter. As for "what they would have told them," probably nothing more than what was in the Frisch-Peierls Report, which essentially said that if you could get a modest amount of pure U-235, you could make a bomb out of it. That is indeed a useful piece of information but it required considerable checking, etc., and it doesn't sound like they pushed it as something very likely.

The real British "intervention" was sending the MAUD Report in the summer of 1941. It was not circulated within the US, and got no immediate obvious response. As a result, Marc Oliphant was sent to follow up on it, and he disclosed its contents to Vannevar Bush, Arthur Compton, Ernest Lawrence, etc. These three in particular "mobilized" to accelerate the American program as a result (changing it from the research-only Uranium Committee into the pilot-production S-1 Committee, which, in turn, eventually became the Manhattan Project).

So the UK did have a very important role in getting the Manhattan Project truly "off the ground" — e.g., out of the doldrums of the Uranium Committee — but it came later, in 1941, not in 1940. As for sources, this is well described in the specific literature on the atomic bomb history (e.g. Hewlett and Anderson's The New World, Rhodes' The Making of the Atomic Bomb) but it is indeed minimized in many other "popular" versions of the Manhattan Project which make it seem like the work started with the Einstein letter. Without the British intervention in 1941 it is unlikely the US would have developed an atomic bomb in time for use in the war (for better or worse).

Here is what Hewlett and Anderson say about atomic bomb content exchanged (258-259):

Uranium had figured in interchange since the days of Tizard. The British mission had barely arrived when Cockcroft asked about fission. Bush first checked with the Navy, then authorized Cockcroft to meet with the Briggs committee [Uranium Committee]. The first of a series of British reports arrived in November, 1940. Conant did not bring up uranium during his talks in England in the spring of 1941, for he knew that it was a special, highly secret phase of the NDRC. Nor did the British raise the question in any pointed way. Hans von Halban told Conant of his heavy-water experiments entirely in terms of a power plant. Professor Frederick Lindemann, the eccentric Christ Church physicist who had become Churchill's most intimate adviser, mentioned atomic power one day at lunch. When Conant expressed his skepticism over its value as a war project, "the Prof" agreed but said that a uranium explosive was barely possible. Conant regarded Lindemann's comments as private and unofficial. The scheme seemed so vague and highly speculative that he did not even report it on his return. Information from England did not influence the American uranium program until the summer of 1941 [the MAUD Committee report].