This is a very interesting question. To continue the answers posted prior to this, I think its fairy safe to say that the first metal screws in Japan were those in connection to the bisen-screw. This was apparently a new technology that proved hard to master for the local blacksmiths tasked with the job.
There are very few types of Japanese matchlocks which uses screws. This is actually true for pretty much every matchlock coming out from South East Asia. The Yonezawa-matchlock mentioned by Piers is the one that stands out. It has a large steel screw securing the trigger-guard to the stock.The same screw continues up and attaches to the bisen, like Ian described. But this type of matchlock also got an iron ramrod, which on it´s tip got a screw-on cap under which a "worm" is to be found. This worm was used to remove duds. A third place you will find threads is found on the hibasami. I have included a picture from my Yonezawa. The threads are clearly visible along with an hand forged nut.
This is very rare to find on matchlocks. This feature is found on the Yonezawa, Seki and so called Kishu-guns.
That the Seki gun is more or less a straight copy from the Yonezawa is not strange. The first Seki-gun was made by an student in Yonezawa that later opened up his own shop near Edo. So that construction is highly influenced by the original Yonezawa-gun which was manufactured 1604, about 15-20 years earlier than the Seki. There are also nuts securing the hibasami on guns made near Negoro/Wakayama in Kii. Unfortunately these are all from the 18th and 19th century. The original Negoro-guns were amongst the earliest matchlocks made after Tanegashima. So it would be fantastic to find an old Negoro-teppo to see if there were any screws fitted on these.
The earliest matchlock I know of with a very similar construction to that of the Yonezawa, with a screw fastened into the bisen, is pre-Sekigahara matchlocks from the Satsuma area. These guns are quite different from the traditional Tanegashima-gun. The origin of these large caliber Satsuma guns are very interesting indeed.
So to wrap this up, I think the first screws seen in Japan were connected to the first guns from Tanehashima and was used to secure the bisen to the barrel. Judging from the few extant matchlocks made prior to 1600, I would guess that screws located on other places on the matchlock might have been introduced during the 1570-90s. Which in the end inspired the makers of the Yonezawa-gun to produce their style in 1604.