In theory (where there are not variants and all's perfectly planned) it works. Optimal sperm production (spermatogenesis) requires temperatures several degrees Celsius below average body temperature. The testes are located outside the body in the scrotum, which has several mechanisms to keep the testes cooler than the rest of the body. Warming the testes to or above body temperature temporarily disrupts spermatogenesis.
Multiple researchers in the 1950s and 60s did proof of concept studies of external heat as a male contraceptive (Robinson 1968, Steinberger 1959, Watanabe 1959). These studies universally showed decreased sperm counts of their human and animal participants, but none tested the methods effectiveness as a contraceptive.
The main evidence for effective use of external heat as a male contraceptive comes from the unpublished works of a Swiss doctor working in India between 1930 and 1950. Dr. Martha Vogeli advised 100s of men on the use of hot sitz baths, and claims that her patients attained 100% effective contraception using this method (1956).
Dr. Vogeli tested baths in a range of temperatures with 9 volunteers to determine the most effective regimen. She determined that 21 consecutive days of 46.5 ºC (116 ºF) sitz baths lasting 45 minutes typically provided 6 months of contraception. However, there was a range of duration of the contraceptive effect from 4 to 8 months just among these 9 men. Dr. Vogeli s volunteers did not continuously monitor the temperature of the bath, but presumably it decreased over the course of the 45 minute treatment (Vogeli 1956).
In the accounts of her work, Dr. Vogeli did not mention a delay in the start of effectiveness. However, heating as a contraceptive method would likely require at least 2 months to become effective. The onset of effectiveness depends on the individual.
Because this method of contraception has had few clinical trials, safety issues are relatively unexplored. Trial participants have reported no effects on libido, mood or physical health. Although no data have been collected for humans, rats in external heat trials experienced a one-third reduction in the volume of their testes 6 weeks after treatment (Steinberger 1959).
And then, there'sthe reversal time and all that. Isn't it easier and safer to buy and use a condom?