The largest conventional builders are the Germans, Danes and Russians (Germany has bought out the swedes), also the French.
Sorry, but the Danes are no major sub builders at all. All subs in danish service are foreign designs (mostly german) bought or build under license.
There is a new sub program in Denmark, the Viking Class, yet it is a trilateral project in cooperation with Norway and Sweden (actually by far the largest scandinavian sub manufacturer, though the Kockums Shipyards now belong to german HDW).
As for the taiwanese sub deal: The current situation seems to be that the US is not able to keep up with the promised provision of the vessels. As was noted before, the US relies on providing a foreign design since they have not build conventional subs for about 50 years.
It was planned to use the german type 212 or 214 design (with the 212 considered the most advanced conventional design in existence), after HDW has been purchased by US One Equiety Partners, yet to prevent technology transfer, the german government has pressed for reintegrating the shipyards back in a german-only consortium, what now has taken place some time ago.
Now Germany refused already in the beginning of the 1990's to deliver subs to Taiwan since there exist very strict rules for exporting german weaponry to conflict zones and this is regularly a big issue of domestic politics, especially among the current government of the social democratic/green coalition (same case with Turkey and their interest in german Leopard 2 MBT), so the chinese factor is but a part in the reasons.
As for France: They also refused to deliver ships some years ago.
France is considered the most serious rival of Germany in the field of conventional submarines. Although they still might change their mind about the deal, it seems unlikely at the moment.
So the last possiblity to get new subs would be Russia with their Kilo class. Yet some meetings about a deal have already taken place and ended in no solution. This may be due to the considerable transactions of military hardware between Russia and China and the influence, that China can use because of this.
Last possiblities I heard of where the purchase of used subs soon to be phased out in other countries. That would be the Gal-Class of Israel (yet it seems that Israel decided to keep these units) or the Spears-class of South Africa (french Daphne-Class). Since both solutions would only result in the purchase of rather old vessels with limited capabilites, it is rather doubtful that Taiwan will take this way.
I dont know about Australia and their willingness to allow a purchase of the Collins class, yet that is pure speculation on my own.
All in all the three most recognized countries/manufacturers are Germany, Russia and France, with Germany having a slight edge in terms of variety and technology over its competitors (though Russia with the large and capable Kilo class is little bit in a field of its own).