Yes, you can use warlock slots to cast any spell you know. It's in the PHB in the section on multiclassing. Page 164:
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
The effect is determined by the slot's level, as always in D&D 5e.
The Healing Light feature gives you a bonus action which basically replaces the Healing Word spell — which you don't get.
You're going to be leaning heavily on that, because your selection of healing spells is limited — the warlock doesn't have any on its normal list (except some necromancy stuff, which only heals you), so all you have are those from the Celestial Expanded Spells feature: Cure Wounds, Revivify, and Lesser and Greater Restoration.
At 6th level, it is probably going to be fine. You're missing the "big guns" like Mass Cure Wounds and Heal, but those aren't available to anyone at this level. And fortunately, you've got bards for the Song of Rest, so your party has short-rest healing benefits available.
I don't think you need to do anything particularly special to "build" this — just avoid focusing on other things that will consume your bonus action for Healing Light. You could look at the Healer feat, but in most cases going for straight Charisma boost is going to do more — it increases the number of dice you can spend at once on Healing Light, and significantly improves all of your non-healing stuff, while the feat just gives you a weak alternative to the Cure Wounds spell you already have. The Acolyte background would give you story-based access to temple services, which might be handy depending on the game and setting — but it sounds like you already have a background in mind.
For a Charisma-based divine caster, you may look instead at the Divine Soul sorcerer — you could generally flavor this to fit many similar story ideas, and you'd have access to the full cleric spell list (and twinned Healing Word is awesome).
Best Answer
If only one class knows/can prepare the spell, it is only associated with that class
The section in the PHB about spellcasting while multiclassing will help with much of your confusion.
When you multiclass as a spellcaster, the rules change slightly with regards to how your spells are prepared/known and cast.
Let us assume that you only know cure wounds from your class in warlock and that you never learned it from bard. In this case, when you cast cure wounds it is always considered to be cast as a warlock spell. What this means is that it will use your warlock's spellcasting modifier and spell save DC and any other qualities that affect the spell you are casting. Also it would qualify for any features that relied on something being a warlock spell but not be able to take advantage of anything requiring it be a bard spell.
However, you share slots and Pact Magic slots between the classes
One place where the spells do overlap is in what slots you can use to cast them.
Thus you can cast cure wounds (still as a warlock) but using any slots that you have available. For example, as a Warlock 1/ Bard 3 you would have 5 first level slots and 2 second level slots available to cast any spell that you know. Which means you could cast that cure wounds using a second level slot if you wanted.