Not exactly.
The club was originally formed as Llansantffraid F.C i.e. a Welsh club, located in the tiny village of Llansantffraid-ym-Mechain and playing in the Welsh system (obviously). It was then re-named TNS, after club sponsors Total Network Solutions, a company based in Oswestry, just a few miles away over the border in England.
TNS then merged with Oswestry Town FC, and the new club started playing at Oswestry Town's ground, making them "English". Except that Oswestry Town had always played in the League of Wales!
Meanwhile, when Total Network Solutions were taken over by BT, the club had to get a new name, eventually choosing The New Saints, partly to retain the initials, but also because Llansantffraid F.C were always known as "The Saints".
The club badge also seemed designed to keep both sets of fans on side:
https://duckduckgo.com/i/def56044.png
Probably best to consider them "Anglo-Welsh", thereby ****ing off fans on both sides of the border!