'Mother tongue' is a useful way to refer to your native language sometimes but 'mother tongue teacher' is definitely wrong.

"A mother tongue teacher." Nosotros no usamos esta frase en ingles.

Hello there, Native English speaker from Texas here! We typically just say native teacher or native speaker. I've only heard the term "mother tongue" from some of my colleagues from India.

You can write "a native English speaking teacher" or a teacher whose mother tongue is English." A fluent English teacher would probably be the best description. Mother tongue implies one language was spoken at home, or that one language is more familiar to a bilingual person. Native speaker implies that the language was spoken from early childhood.