Which faction that will support which group depends entirely on the internal politics of those nations. It's also entirely likely the various nations will vary their support between mages and templars in different nations.
For instance, it's fairly likely that the Tevinters will support the mages in the Anderfels, Nevarra and Antiva. Empowering them not out of the goodness of their hearts but to prepare for future expansions in those directions, alternatively influencing a more pro-mage stance in order to foster less hostile tensions along the border to allow them to focus more resources on the qunari. Similarily, chances are the Tevinter mages will support both sides in Orlais. Because the fiercer the fighting is in Orlais, the less will Orlais be able to oppose Tevinter plans.
In Orlais itself, the various sides in the civil war will be courting all three sides: mages, chantry and templars. They all need the edges the mages can provide, but none of them will want to cede some of their power to the mages. Both sides need the chantry to legitimize them, so that is extremely valuable. But whichever side that provides the best chances to secure the throne will be supported. Until which one is certain, both Celene and Gaspard will play all sides in the conflict.
Nevarra is expansionistic and will look towards the side that provides the most benefit from that. It will likely support the side that that weakens Orlais the most
in Orlais but inside it's own borders it will focus on combating the side that risk destabilizing Nevarra the most. So they might very well support Orlesian mages but oppose them at home. The Nevarrans might also support the templars at home and in the Free Marches partially to limit Tevinter influence and in order to secure control over Kirkwall.
I think Anderfels are going to stay fairly loyal. Unlike the other nations, they seem to have a lot less dissatisfaction among the mages (seeing that the inetrnational head of the Loyalists was the Hossberg First Enchanter... which is a position that is elected), likely a result of the Grey Wardens providing a steady alternative for all mages dissatisfied with circle life and Anderfels being the only nation in the world where the GWs can recruit several mages at once. They're also the nation that has the largest habitable border towards Tevinter and Hossberg will seek to resist their influence. So templars with empowered Grey Wardens I think, or if the templars seek to impose itself on the grey wardens as well: Chantry.
In Antiva it will depend on who can afford the most Crows

Rivain and Ferelden are more tricky. The rivaini being split between it's natives, it's andrastians and it's viddarthari. They might steer towards templar/chantry since Darsmuid has been annulled (and with it most anti-Chantry mages) and the risk that qunari might just use the excuse to "put things right". Chances are that if fighting erupts in Rivain it'll lead to religious conflict that makes Orlais look peaceful.
In Ferelden it depends entirely on what the Bannorn wants. Alistair is a popular king, and since the Bannorn wants their higher-ups to meddle as little in their affairs as possible it's unlikely that an unhardened Alistair is a powerful king. Anora or a hardened Alistair has probably more power, but overall I think Ferelden's true power lie with the Guerrins (only rivalled by the Couslands). The Guerrins seems to have ties with Celene and they have been "burned" by mages, so that speaks in favour of Chantry. Getting rid of Orlesian templars will be tempting for them though, especially if the Warden was a mage, and if Orlais is speaking in favour of expansion and forcefully pacifying mages then they're probably going to protect their own mages.
Basically, Ferelden could go either way. But is probably going to focus more on staving off war with Orlais than actively involving themselves if they can.