2016 HCM uses fixed values of 4.5 s and 2.5 s for critical and follow-up headway that must be based on some field data. Do we know if these were verified based on the latest research that HCM 2016 used? What are the background studies that led to these values?

I believe you're referring to the gap acceptance parameters for permitted left turns provided in Chapter 31. The values were not updated in the HCM 6th Edition (HCM 2016) because the signalized intersection method was not updated in this edition. We asked the chapter author about the source of the values and received the following answer: "The values of 4.5 s and 2.5 s for critical and follow-up headway have been in the HCM Signals chapter since 1985. I believe that the origin/derivation of these values is described in the following publication: Fambro, D., C. Messer, and D. Andersen. (1977). “Estimation of Unprotected Left-Turn Capacity at Signalized Intersections.” TRR 644, Transportation Research Board, p. 113-119."