Confusion may arise because both "that said" and "said that" are common phrases; however, their meaning is very different here.
The first example, "A report yesterday that said British unemployment would continue to rise." is not a complete sentence but a phrase describing the report. That is, both "yesterday" and "that said British unemployment would continue to rise" modify "report"; there is no verb to follow. Other examples of restrictive relative clauses following a noun are
A report that addresses recent events
A report that lacks vital information
A report that looks like a five-year-old made it
Your rewording ""A report yesterday said that British unemployment would continue to rise." instead forms a complete sentence, as in the following:
A report said that the bridge is unstable.
A report noted that crime is decreasing.
A report argued that the problem is insignificant.
Here, "that" acts as a conjunction introducing the object of the verb.