Praise be to Allah.
The point of the greeting of salaam and the adhkaar that are prescribed is the meanings, aims and impact, and undoubtedly that may be achieved by means of a language other than Arabic. Hence the scholars ruled that saying these things in Arabic is not a must.
An-Nawawi (may Allah have mercy on him) said:
With regard to giving the greeting of salaam in a foreign [non-Arabic] language, there are three opinions, the third of which is that if a person is able to say it in Arabic, it is not valid in another language.
I [an-Nawawi] say: the correct view is that it is valid to give the greeting in a foreign language, if the other person will understand it, regardless of whether or not he is able to speak Arabic, and the response must be given, because it is still regarded as a greeting and prayer for peace. And Allah knows best.
End quote from Rawdat at-Taalibeen (10/230).
They also said that this is applicable when mentioning the name of Allah at the time of slaughter.
Ibn Qudaamah (may Allah have mercy on him) said:
If he mentions the name of Allah, may He be exalted, in a language other than Arabic, it is valid even if he is able to speak Arabic, because the aim is to mention the name of Allah, and that is possible in all languages. This is in contrast to saying the takbeer (“Allahu akbar”) in the prayer, where the aim is to say the exact wording (in Arabic).
End quote from al-Mughni (13/260).
But what should be done, if the speaker and the one whom he is addressing both know Arabic, is that the greeting of salaam and the response should both be given in Arabic. The scholars stated that for an Arab to speak any language other than Arabic unnecessarily is makrooh (disliked).
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Ash-Shaafa‘i said: … The language that Allah, may He be glorified and exalted, chose is the language of the Arabs. He revealed His holy Book in Arabic, and made it the language of the last of His prophets, Muhammad (blessings and peace of Allah be upon him). For this reason we say that anyone who is able to learn Arabic should learn it, because it is the preferred language, even though it is not haraam for anyone to speak any other language.
Ash-Shaafa‘i regarded it as makrooh for the one who knows Arabic to say “Bismillah (in the name of Allah)” in any other language, or to speak Arabic mixed with another language. What the leading scholars said was also narrated from the Sahaabah and Taabi‘een.
End quote from Iqtidaa’ as-Siraat al-Mustaqeem (1/521-522).
And Allah knows best.