Yes, a native speaker would understand with no problem. You don't have to be able to make out every single sound and syllable of a word in order to 'hear' it. Your mind naturally fills in whatever your eyes and ears don't perceive. Listening comprehension is about understanding, and not really about how well your ears work.
In every uttered sentence, there are only a few important elements that you need to pick up in order to make sense of the whole utterance. The word remember has three syllables, but only the middle one is stressed, and so this is where the crucial part of the meaning of this word is transmitted, so as long as a speaker pronounces 'mem' or even 'me-' (as the man does here), it's probably enough. At the beginning of his utterance, you can just about pick up a /dz/ consonant representing the words Do you, which tells the woman this is a question. The rising final intonation of the whole thing also makes it very clear this is a question that needs to be answered. Notice how he makes much more of an effort to pronounce the name Tommy Sheridan, as he understands that this word is loaded with meaning and must therefore be clearly pronounced.
So I think we essentially have four internal elements here that carry the entire message of the sentence:
/dz/ = Do you
/'me/ = remember
/'tɒmɪ'ʃerɪdən/ = Tommy Sheridan
rising intonation = ?
There is a fifth element too, so to speak, which is the context of the conversation itself. Any utterance really only makes complete sense in the context of the words, sentences, and paragraphs around it.
