My true feeling is that the expression 「~~も[何]{なに}もあったものではない」 should be memorized (and actively used) as a common idiom than be analyzed grammatically.
It is an expression of "total" negation. "There is no ~~ whatsoever!"
「~~も何も」 lumps together things that are like or related to "~~". See http://kotobank.jp/word/%E4%BD%95%E3%82%82?dic=daijisen&oid=13802800
「あった」: Do not take this to be the past tense. Rather, take it to be in the hypothetical form because with this phrase, you want to say "~~ could not exist." You are only stating your opinion; You are not stating a fact. What I mean by that is that 「あったものではない」 is not an expression about something that actually existed, exists or will/might exist. In that sense, it is tense-free to begin with.
A better-known expression for J-learners would be 「~~した方{ほう}がいい」, which is also an expression for stating an opinion rather than a fact. People often ask "Why use the past-tense た?" in it, but again, that is NOT the past-tense た. The speaker is only giving advice; It is only his opinion. Just like 「あったものではない」, this is not an expression about someone actually doing something in the past, present or future. It is "tenseless".
「もの」 here is a nominalizer with a special function to express one's judgement emphatically. See Definition 5-ア in http://kotobank.jp/word/%E7%89%A9?dic=daijisen&oid=18293900