"What month is the week after next?"
げつ means "moon" or "month" but is seldom used alone. It is mainly used to form words such as 月曜日（げつようび） "Monday" 満月（まんげつ） "full moon" 月給（げっきゅう） "monthly salary."
がつ cannot be used alone and is used as a suffix to mean "the month of." 正月（しょうがつ） "the new year (literally the first month)" 生年月日（せいねんがっぴ） "date of birth (not really used as a suffix though)"
As far as I have seen, when 月 is used with other kanji to mean month, the reading is げつ when the month is not specified (e.g. 先月, last month) and がつ when talking about a specific month (e.g. in this case 何月, which is asking for the name of a month as an answer). When by itself, it's pronounced つき.
Jisho.org has an entry for なんがつ (https://jisho.org/search/%E3%81%AA%E3%82%93%E3%81%8C%E3%81%A4) but not for なんげつ (https://jisho.org/search/%E3%81%AA%E3%82%93%E3%81%92%E3%81%A4).
the 12 months as in january, feb, and so on, 月 is read as gatsu. but the words, next month, this month, last month, the 月 is read as getsu. but in "every month" 月 is read as tsuki. Tsuki is the word for moon. January = 1月 いちがつ. February=2月 にがつ. next month=来月 らいげつ. this month=今月 こんげつ. every month＝毎月 まいつき
The unmarked word order is <topic (は)> <time> <everything else> <verb> <sentence particle (か/ね/よ)>. The objct (を) is often close to the verb, but otherwise the order of "everything else" isn't really specified (except that things that have to do with each other tend to be placed together, of course). See https://8020japanese.com/japanese-word-order/
3 problems with your answer.
何月 (asking for which month) is "nangatsu" not "nangetsu."
You must use が if the question word is before the particle and は if the question word is after. So "wa" is not correct because "nangatsu" is at the beginning.
It is unnatural to put the question word in front. There are no inversions in Japanese like English. "What month is it?" is an inversion. The answer is for example "It is January." However in Japanese you do not say 何月が今月ですか but instead say 今月は何月ですか and the answer is 今月は一月です。Subject remains subject without inverting it to the end of the sentence.