You're going to get to 120 in probably about 7-8 hours on your first play. After that i'd wager about 4-5 with looms, maybe even faster.
Keeping this in consideration, how long does it take to level to 110 BFA?
Took me 2 Days 17 Hours played to level 1-110 as an Arms Warrior after the Pre-Patch (Before they nerfed 60-80 as well) by just questing and the occasional dungeon. I'd just boost to 110 though since you get a free boost with BFA.
Furthermore, how do you unlock Argus on boosted 110?
In order to get to Argus you must complete the intro quest line that starts with "The Hand of Fate". The following article outlines the steps to qualify for that quest. If you do, Archmage Khadgar should have the quest for you in the Violet Citadel in Dalaran. Make sure you have trivial quest tracking on.
How fast can you level to 120?
If you're leveling with full heirlooms (~45% bonus xp), war mode on (10% for horde, 30% for alliance), with a plan for what path to take to level and a few quality of life tricks, you can get to 110 in about 48 hours, and an additional ~6 hours to 120.
Is it worth using heirlooms to 120?
Upgrading Heirlooms to 120 - not worth it! I made more XP per hour questing with azerite gear compared to my heirloom gear. Azerite gear provides substantial damage increase across the board and the stats are nearly twice the amount compared to heirloom gear.