Must-have mons for great and ultra leagues?

In addition to the ones you mentioned, I'd also say the following are top tier picks.

Great League:
• Azumarill (bubble, play rough, ice beam)
• Meganium (vine whip, frenzy plant, earthquake)
• Umbreon (snarl, dark pulse, last resort)

Ultra League:
• Kingdra (dragon breath, outrage, and hydro pump or blizzard)
• Charizard (fire spin, blast burn, dragon claw)

I focused on the 1500 bracket so far. Besides the two you mentioned, my favorites are:

Whiscash (Mb+Bl)
Azumarill (PR+IB)
Skarmory (SA+SW)
Alolan Muk (DP+GS)
Alolan Sandslash (GB+Bl or Bd)

I did not test Flygon yet, but I assume it is a good choice as well.

Great:
Cresselia
Flygon

I'd recommend the following from my experience:

- Piloswine (Powder Snow + Avalanche / Stone Edge) - Great and Ultra
- Magneton (Spark + Zap Cannon / Magnet Bomb) - Ultra
- Victreebel (Razor Leaf + Leaf Blade / Sludge Bomb) - Ultra
- Swampert (Mud Shot + Surf) - Ultra and Master
- Charizard (Fire Spin + Blast Burn / Dragon Claw) - Great and Ultra
- Blastoise (Water Gun / Bite + Hydro Cannon / Flash Cannon or Ice Beam) - Great and Ultra
- Scyther (Air Slash / Fury Cutter + Aerial Ace / X-Scissor) - Great
- Giratina (Dragon Breath + Ancient Power / Dragon Claw / Shadow Sneak) - Ultra
- Linoone (Shadow Claw + Thunder / Dig / Grass Knot) - Great

Ultra League
Flygon Mud Shot/Dragon Claw
Sceptile Fury Cutter/Leaf Blade
Gligar(Great League)/Gliscor Fury Cutter/Aerial Ace

Ultra:
Latios
Typhlosion with SC is a stealth Psychic killer, and Metagross doesn't stand a chance against it. More useful than dual fire Charizard with the same stats and an extra set of weaknesses.
Flygon
Giratina
Swampert
Torterra

Great:
Lucario, BP/SB is cruel
Flygon
Azumarill
Kingdra

Note that no set of three is unbeatable; I find several of the ones above my team is particular well equipped to handle whereas my fourth team (created, not in terms of strength) would be poorly equipped to deal with Piloswine in particular.

Also note the prevalence of dual-types with off-type charge moves on my personal list.

This isn't the most scientific analysis, but I've found the pokegenie "duel" stat has correlated well with first-hand experience in these battles and correlates well with what many of you have had to say. This number incorporates the damage output and tankiness.

I had a half hour to kill, so I ran all the pokemon mentioned in this thread and a few others (in the Ultra category) through pokegenie with their CPs maxed up to the 2500 limit. This is what I got. I showed where a level 30 Blastoise would sit as well, because I for one am not going to spend the dust to max one out.

Take it with a grain of salt if you will:

Blastoise (level 40, HC): 50
Giratina (dragon moves): 48
Muk (regular and alolan, PJ/GS): 43
Metagross (MM): 42
Kingdra (dragon moves): 41
Alolan Muk (dark moves): 41
Charizard (BB): 40
Blastoise (Level 30, HC): 40
Giratina (ghost moves): 39
Swampert (water moves): 39
Flygon (dragon moves): 39
Kingdra (water moves): 39
Typhlosion (BB): 39
Sceptile (FC/LB): 37
Victreebell (level 40, RL/LB): 36
Sceptile (RL/LB): 36
Victreebell (poison moves): 35
Latios (DB/DC): 35
Jolteon (TS/TB): 34
Swampert (ground moves): 34
Mewtwo (C/SB): 34
Flygon (ground moves): 31
Magnezone (Level 40, S/DC): 31
Magnezone (Level 40, S/MB): 30

The problem is (which is why I'm waiting for someone to do the real math) that people have been reporting the punching bags (Chansey and Blissey) haven't been that great, but their numbers are very high for the "duel" stat of the great and ultra. For example, Blissey would top this list with a number in the upper 60s for P/HB.