A pronunciation like [sũːmʊs] for summus in Classical Latin seems rather unlikely considering that the Italian reflex is sommo [somːo] and not [suːmo].
The letter M in Latin corresponded to the nasal consonant [m] in any position but word-final (and excluding compound words, which I will discuss in the section below)*. So sumus, summus, amplecti, sumpsisti are something like [ˈsʊmʊs], [ˈsʊmːʊs], [amˈplɛktiː], [suːmpˈsɪstiː]. It seems plausible that the vowels preceding the nasal consonant [m] in these words were pronounced with some amount of phonetic nasalization, but that kind of "nasal vowel" would not be phonologically distinct from a regular Latin vowel phoneme, as the presence of nasalization here would be entirely predictable from the phonological context (as is the case for vowel nasalization in present-day English, which isn't normally marked in phonetic transcriptions). I don't know of any actual evidence that these vowels were nasalized.
The possible existence of nasal(ized) vowels in word-final position, or before voiceless fricatives, is notable because we have evidence that the nasal consonants that originally conditioned the nasality of the vowels may have been lost in these two contexts, with "compensatory" lengthening of the vowel. We have clear evidence of the lengthening of vowels before NF and NS, and fairly clear evidence of the possible loss of the consonantal value of N before S, and of M in word-final position before a vowel. The existence of distinctively nasal long vowels in Latin (as opposed to non-nasal long vowels, or vowel + nasal consonant sequences) is mostly an inference, not something that we have clear evidence for (and so, as Alex B.'s answer mentions, it is a bit controversial).
*The pronunciation of M inside compounds might be a bit unclear in some cases
The pronunciation of certain compound words where the first element ends in m and the second element starts with a vowel, e.g. circumeo, is apparently doubtful. Andras Cser's "Aspects of the Phonology and
Morphology of Classical Latin" (2016) has a section on circum- (188.8.131.52.). In some forms of circumeo, it was common to not write the letter M (e.g. circuit), which could be interpreted as an indication that nasality was totally lost in these forms.
An M that was originally in word-final position is assumed to have been pronounced as a nasal consonant before a plosive or nasal consonant, with assimilation to the place of the following consonant. So [m] before [p b m], [n] before [t d n], and [ŋ] before [k g]. For some words, we have evidence for these changes in the form of a spelling with N: thus eam + -dem > eandem.
This kind of place assimilation is thought to have applied even when the consonant remained written as M. So before the enclitic -que, M was probably pronounced [ŋ] (making the form "uirumque" something like [wɪˈrʊŋkwɛ]), and before the enclitic -ne, M was probably pronounced as [n] (making iamne [ˈjanːɛ] and mātremne [maːˈtrɛnːɛ]).