Go old school, and encourage the players to hire henchmen. Noncombatants (unless forced) who can tend to wounds after the battle. This works especially well if one of the PCs has the Noble background: one of the retainers can be an acolyte with herbalism skill and minor clerical powers (spare the dying and healing word).
The Warlock is another option. If he takes a single level of cleric, he'll be able to cast any (1st level prepared) cleric spell using his warlock spell slots--and those slots are always at max level. For example, when that Clr 1/Wlk 5 uses a Warlock slot to cast Healing Word, it'll be cast at 3rd level. He can cast two of those per hour. So, with a couple hours' rest he should be able to patch the whole party.
Keep in mind that this party is not just missing a healer, it's missing spell diversity. Key spells like Protection from Evil, Dispel Magic, Zone of Truth, Teleport. Even if you cover HP replenishment with NPCs or items, they're going to lack some important utility spells.
There are two other ways to patch that deficiency: Ritual Caster feat, and/or the magic (and magic-like) paths of Fighter (Eldritch Knight) and Monk (Way of Shadow or of Four Elements). The Warlock could cover the rituals with Book of Ancient Secrets.
Alternatively, you could have them rely solely on spending their own HD. 5e does have a self-heal mechanism. And the fighter and monk (Open Hand) both have self-heal class features.
Feats can also help. Encourage the fighter to take Heavy Armor Master for DR, and the Warlock to take Inspiring Leader for a reliable source of a significant number of temp hit points. All three could potentially take Defensive Duelist for on-demand AC. Consider giving everyone a bonus feat, but limit the choices to the above (plus Healer).
Or, you could sidestep the issue. Just say that after a fight, everyone heals a certain amount. In previous editions I had some success with healing everyone up to the max of whatever threshold they were in. So if they were bloodied, they healed enough to put them at exactly 1/2 max HP. If they weren't bloodied, they healed to full. There's no bloodied state in 5e, but you could certainly add it (and the oft-mentioned Staggered for 1/4 max HP).