Let's take amōrum and armōrum as a minimal pair. Both have a short vowel in the first syllable. The difference is that in amōrum, the syllable ends in the short vowel, while in armōrum, the short vowel is followed by the consonant r. Any syllable that does not end in a short vowel is heavy, so the first syllable of armōrum is heavy.
Without doing anything special, just pronouncing both the short vowel a and the consonant r in sequence (/ar/) should take more time than just pronouncing the short vowel a by itself. So as Joonas commented, it's not necessary to think of things in terms of lengthening either the vowel or the consonant in this context.
When you have a long vowel followed by a consonant in the same syllable, as in ēmptus, you just pronounce the vowel as long, and then pronounce the following consonants.
It's not entirely clear why, but linguists have observed that in many languages consonants at the beginning of a syllable don't seem to count for the weight of the syllable (even though consonants in this position also take time to pronounce).
In fact, there are ancient grammarians who noticed and mentioned the fact that some long syllables are actually longer in duration than others. E. H. Sturtevant, in "Syllabification and Syllabic Quantity in Greek and Latin" (Transactions and Proceedings of the American Philological Association, Vol. 53 (1922)) cites Dionysius of Halicarnassus, who notes that the first syllable of "ὁδός" is short and the first syllable of "στρόφος" is also short in terms of meter, even though the initial syllable of the latter word has three additional consonants at the start; and that the word σπλήν is longer than the vowel η by itself, even though the vowel on its own would scan as a long syllable. This is about Greek, but these kinds of observations surely applied also to Latin quantitative poetry (which owed a large debt to Greek).
Thus, it is not really problematic that a word like dēmptum likely had a longer duration than a word like dēmum, despite both having the same value in poetry (two heavy syllables).
Despite the appeal and intuitive helpfulness of the idea, the scansion of heavy and light syllables in Latin meter probably did not correspond exactly to how long it naturally took for Latin speakers to pronounce the syllables. (I'm disagreeing somewhat here with the article by Sturtevant that I cited here; you can read it for a different viewpoint: he rejects conventional syllable division and thinks duration was important to Greek and Latin meter.) Natural speech does not have a regular tempo like a metronome, and setting aside the matter of phonemic length, different consonants and vowel sounds (e.g. /a/ vs. /i/) take different amounts of time to say (mentioned in this Language Log post by Mark Liberman: Slicing the syllabic bologna, May 5, 2008). Some more complications commonly seen in regard to phonetic duration are discussed on this web page: Temporal Structure (Louis Goldstein, USC)
The heavy final syllable of "Amor" in "omnia uincit Amor et nos cedamus Amori" is a (well known) metrical abnormality: we would expect to find a light syllable in this context (since the /o/ is normally short, and the /r/ normally would be syllabified with the following vowel). In this line and a number of others, we instead find a word-final syllable scanned as heavy. There is no consensus on the reasons for this kind of scansion, or what was going on phonetically when an ancient Roman read this kind of line. It's simplest to just remember that this kind of thing can occur in hexameter, rather than trying to understand why or how it was pronounced, since it seems nobody alive knows for sure.
However, if you want to go into more depth on that topic, you can read the arguments that have been made in journals for various interpretations (among the hypotheses I've seen are a) that it comes from archaic pronunciations that had a long vowel in this position, b) that it comes from a pause that prevented resyllabification of a word-final consonant, and c) that it was a license allowed because of the low functional load of vowel length contrasts before a word-final consonant). Here is one paper about the phenomenon: ‘Irrational Lengthening’ in Virgil, Mnemosyne, by Rupert Thompson and Nicholas Zair, published online 28 Feb 2020