The following works for all but necroa and Neurax (and with minor adjustments for some at the bottom). Depending on plague type, you should expect 250-260 days if all goes well; most of my records are well below that. Bottlenecks: Greenland and Caribbean typically; if they don't infect, try again.
(0) start in Saudi Arabia (optional; any hot non-rich country in Asia with airports and ports will do)
(1) make sure to get ATP boost, Darwinist, and Sympto-stasis
(2) get all level 1 and 2 symptoms other than paranoia, plus, from level 3, skin lesions and diarhoae. Rush skin lesions first, and get the non-infective ones last (insomnia, ...).
(3) get all 5 from water and air transmissions.
(4) get drug resistance 1 and 2
(5) get both heat resistance, then both cold resistance, and optionally the 5th temperature resistance. More aggressive: mix in symptoms like dysentery, systemic infection, or necrosis.
(6) get systemic infection and maybe necrosis or dysentery, then wait for every country to be infected. As soon as that happens, get all lethal symptoms, starting with total organ failure; and with paralysis, insanity, and coma last. You can try to be more aggressive here if you have all 5 temperature resistances as with all the resistances, you should infect faster than you kill (exception: very early on in New Zealand). Every now and then, you'll not manage to infect everyone - just try again.
Nano: take all free symptoms as soon as available (for paralysis and coma, not before cure is at 30-40% though)
Parasite: in (3), get only the first 4; in (5), don't get the last temperature resistance. We also need to pump lethality early, so get necrosis (and maybe 1 or 2 more infectious and lethal symptoms) before the 2nd level resistances
Virus: taking the 3 mutations sometime during (2) (but after skin lesions) might speed you up (you might also go too fast)
Fungus: get patho-stasis instead, and start the game by using all spore bursts, then get 3 symptoms up to skin lesion, then (3), (4), (5), then necrosis. After All countries are infected, push for total organ failure and the other lethal symptoms. My best run was 237 days, but this only gives me 4 biohazards. There used to be an exploit allowing for faster times
Bioweapon: my best is 228, which only gives you 4 biohazards
Neurax: guide at Yo It's Spicy works; except get resistances earlier
Necroa: seems impossible to get 5 biohazards
Edit: I keep updating this if I find slight improvements