Are there some concrete figures for the stage-2 contract? Stage-1 contract was known to be "in excess of £100million" for example.Around 150 afaik. It actually isn't that much considering there are thousands of engineers that are directly working on the programme at TAI. Their main role seems to be consultancy.
Also, if true, the "thousands of engineers" you've mentioned is probably the number of engineers involved in the entire TF-X ecosystem, from TAI that is responsible for the aircraft itself and the system integration, down to the subcontractors of subcontractors that are responsible for the smallest parts. Based on educated guess (from previous fighter jet development examples), the total number of engineers who are directly involved in the project (main contractor and 1st line sub contractors contracted directly to TAI and SSB) would rather range from a few hundreads to a thousand and a half.