母 *mə- seems to be a morphologically atrophied element originally attached to the stressed root 猴 *go (< Baxter-Sagart's reconstruction *mə-ɡˁo 母猴). 猴 always came with 母 and never stood alone during the Pre-Qin era (e.g., 《呂氏春秋》狗似玃，玃似母猴，母猴似人; 《韓非子》 以棘刺之端為母猴者). By the time of the Han Dynasty, the first syllable was dropped in the Central Plains, but it was preserved in the Chu region, taking the form of 沐猴 *mokgwa (陸璣云：猱，獼猴也。楚人謂之沐猴). The Old Chinese 母猴 was borrowed by other Chinese dialects, in which the originally meaningless first syllable was reanaylzed as an independent word with a lexical identity such as "big" (母音轉爲馬, 馬之言大也 ==> 馬猴) and "mother" (方言呼母曰㜷, hence 母音轉爲彌 ==> 獼猴). The timing of these changes can be inferred from the following: (1) 獼 was not attested in 《说文解字》, and (2) 三國吳 陸璣 defined 猱 as 獼猴. Also, (3) 馬's pronunciation in the Han Dynasty has been reconstructed as *mrǟ́ and in the Early Postclassic Period as *mạ̄́, which is a better match with 母 *mǝ̄́. To sum up, we can assume: Early Old Chinese *mgo "monkey" > Late Old Chinese *mə-ɡo > Han Dynasty *(mə) ɡwa > Early Postclassic *(mV) gōw.
馬's association with "monkey" could be a result of linguistic interference (e.g., Austroasiatic: Proto-Vietic *k-mah, Chứt mah, Alak məəw "monkey"). 馬留/馬流 also has cognates in the proto-forms of neighboring languages (e.g., Proto-Lolo-Burmese *mhlukx "monkey"; Zhuang 壯語 maxlaeuz; 黎语 *mlok). 狖 "a type of ape" was attested in the Western Han but absent in 《说文解字》, an indication that it was a newly introduced word. It was probaby pronounced as *Łuh (余救切), whose latteral fricative mirrors -hl- in Proto-Lolo-Burmese *mhlukx. That is to say, 馬留/馬流 is possibly a loanword.
Of course, based on 黎语 *mlok,Proto-Lolo-Burmese *mhlukx, Old Chinese *mgo, and other proto-languages, we may reconstruct something like *mluko, which however would not be called "Chinese" and therefore will not be discussed here.