The one remaining detail is the "Bishop's bird stump", a large piece of Victorian bric-a-brac. (The exact nature of this item isn't revealed until much later in the book.) As the story begins, a team of Oxford historians is sent to Coventry just as the crucial air raid begins to determine whether the "bird stump" was in fact in the cathedral at the time.
Complicating the effort is the fact that the British knew about the Coventry raid in advance, thanks to the decoding of the Nazi Enigma machine-encoded message by a similar machine in the possession of the British. The knowledge was withheld because the German High Command would have changed the code if they had suspected that it had been broken. The code-breakers were able to supply valuable intelligence later in the war, so anything that compromised the secret, including an interloper from the future, might change the course of history. At first it seems that the paradox is related to the Victorian era, but it relates to the Enigma code.
The protagonist, Ned Henry, is a specialist in 20th century history, assigned to search for the Bishop's bird stump. He has made so many jumps into the 1940s so quickly that he has developed "time-lag", the time-travel-induced form of jet lag, and must recuperate before he returns to work. There is, unfortunately, an emergency in progress. A historian sent to the Victorian era has returned bringing something from Victorian England with her which the historians believe may rip time itself apart if it isn't promptly returned...and Ned, who knows virtually nothing about the 19th century, is the only one available to return it. (Theoretically, nothing may be brought through the time machine in either direction as it might cause time to unravel, and safeguards have been put in place in order to prevent significant objects making the journey.)
Unfortunately, Ned is too time-lagged to recall his destination, much less remember what it is that he is supposed to return. He meets one Terence St. Trewes, a besotted young Oxford undergraduate, and loans him the money to hire a boat for a trip on the River Thames from Oxford down to Muchings End, where Terence hopes to meet his love, Tocelyn "Tossie" Mering. Ned, Terence, Cyril the bulldog and an Oxford don Professor Peddick travel down the Thames, navigating locks, beautiful scenery, crowds of languid boaters in no hurry to get anywhere, and the party of one Jerome K. Jerome (to say nothing of the dog, Montmorency).
Fortunately, Ned's contact in Muchings End recognizes him when he arrives, and identifies herself: she is a young woman named Verity Kindle, who is pretending to be Tossie's cousin. Lady Schrapnell sent Verity to read Tossie's diary, as Tossie (an ancestor of Lady Schrapnell) had written about a life-changing event involving the bird stump at the church which was to become Coventry Cathedral, which had caused her to elope with a mysterious "Mr. C" to America. It is only at this point that Ned learns the nature of the object he is to return: Tossie's pet cat, Princess Arjumand. (Cats are extinct in 2057.)
However, returning the cat didn't clear up the time disruption, as people attempting to visit Coventry during the air raid are still missing their target--they are either in the right place at the wrong time or the right time but miles from the target. Have they changed history by bringing Terence to Tossie? What will happen to them if Lady Schrapnell is never born, to say nothing of Terence not marrying Peddick's niece Maud, and thus not becoming the grandfather of an RAF pilot who bombs Berlin and goads Hitler into bombing London and Coventry?
The solution involves the wisdom of Sherlock Holmes, the methods of Hercule Poirot, and the style of Lord Peter Wimsey. In the meantime, Ned, Verity and their colleagues have to deal with packs of dogs guarding the marrows, hostile theatrical costumers dragooned into operating the time machine, phony spiritualists, kittens, abstruse mathematics, the Battle of Waterloo, the unalterable fact that the butler did it (they always do), the Coventry Ladies Altar Guild, more dogs, and a crime which was committed before anyone realized it was against the law.