Four Aasimar walk into a tavern...
Because the use of spells are out, there isn't too much going on. I'll quickly go through each of the classes, all of whom get a feat which I'll cover afterwards. I'll also assume everyone have a Charisma modifier of +3 and that noone other than party members are available or relevant to heal.
Paladin: All get a pool of 30 hit points to dish which is restored on long rests. In addition, by choosing the Crown oath (SCAG) you can restore 1d6 + Charisma modifier to any number of creatures, once per short rest.
Monk: Choosing the Open Hand tradition, monks can heal themselves 3×Level=18 per long rest.
Cleric: The life domain channel divinity restores 5×Level=30 creatures and at 6th level it can be used twice between short rests.
Warlock: Celestial warlock gets a pool of (1+Level)d6=7d6 healing. Also, they should choose the Pact of the Chain so they qualify for Gift of the Ever-Living Ones (XGtE) which will maximize healing the Warlock gets. (Not included in calculations, because it turns them into a mess.)
This is the non-spell healing available to those classes beyond the expenditure of their hit dice. Then, we have their feats. There is a tiny set of feats which grant healing, however Magic Initiate would do it through a spell, so we're left with:
Healer: Which by expending a 5 cp use of a healer's kit can have a creature regain 1d6 + 4 + number of hit dice = 1d6 + 10 hitpoints. Each creature can only benefit from this once per short rest.
Inspiring leader: Not healing per say, but lets you hand out temporary hit points (which does much the same) to up to six creatures.
Martial Adept: in much the same vain, by choosing this feat and the Rally manouver, you can have one creature gain 1d6 + your Cha mod temporary hitpoints once per short rest.
Note that multiple copies of Healer and Inspiring Leader don't work together, so between your 4 adventurers you should have 1 Healer, 1 Inspiring Leader, and 2 Martial Adept: Rallys. Do note that temporary hit points don't stack, so wait for the old ones to be used up before applying new ones.
If everyone chooses Aasimar as their race (Volo's) they get the Healing Hands feature allowing them to restore hit points equal to level, once per long rest.
To summarise, we get
- 30+18+4×6 = 72 hit points restored per long rest
- 60+7d6+4×(1d6+10)+4×(1d6+3) per short rest
- 4×9+2×(1d6+3) temporary hit points per short rest