Quoting the 50th anniversary page on the topic,
In parallel to the Meg project it was decided to build a relatively small and economic computer. When design was started in 1952 it was clear that the project could provide valuable experience in the use of the recently introduced transistors. It was built even though the germanium point transistors were more unreliable than valves, because semi-conductors held out the promise of lower power consumption, higher operating speeds, smaller size and greater reliability in the future.
A prototype was completed in 1953, and a full-size version in 1955; this was then reproduced by Metropolitan Vickers (replacing all the germanium point transistors with more reliable transistors, and using printed circuit boards), but never sold.
I’m not sure the transistor computer was really intended to solve problems; I suspect its aim was rather to identify the problems which would need to be solved in order to produce a smaller, cheaper, and ultimately, faster computer, using transistors. After all, this was the first transistor-based computer, so no one knew what it would be like to build a computer using transistors. Quoting the same source:
The most important impact of the Transistor Computer was it provided early experience in transistor circuit techniques.
Simon H. Lavington’s Early British Computers puts it this way:
Kilburn’s group at Manchester University built a small transistor research computer to gain some experience with the new devices.
The project certainly contributed to subsequent transistor-based computers built at Manchester University, including the famous Atlas, which did have a problem statement.
The design was led by Richard Grimsdale, who gave a talk on the subject at the Computer History Museum in 2000; this was filmed and is available on Youtube.