I was thinking about some ways to tweak the rules and make my homebrew world seem more unique and fit better what I had imagined, since I created it before I started playing d&d. One of the ways was to add the option for arcane casters to use the spell point rule, while divine casters would have to keep the normal spell slots

By the way, I'm defining arcane and divine spellcasting by what's in the "The Weave of Magic" section from the PHB. I was told it's in page 120. Warlocks are already unnafected by the spell point rule by standard, so disconsider them when I speak of arcane casters

How should it look if a wizard multiclassed into a cleric, for example? Normally their spell slots would just level up, and by the spell point rule they would too, but since one class has access to it and the other don't, how many spell slots would he get as a cleric? Should they be able to cast their cleric spells with both their spell slots and spell points?

If it gets too complicated, I'll probably just allow divine casters to use spell points in the first place, but the mechanical difference feels very important to me. Magic works quite differently in the world that I created than it does in lore-friendly d&d.

For reference, here's the Spell Point variant: https://www.dndbeyond.com/sources/dmg/dungeon-masters-workshop#CreatingNewCharacterOptions

And the Weave of Magic blocktext is in this section: https://www.dndbeyond.com/sources/basic-rules/spellcasting#Targets

Personally, I'd keep the two systems separate. So if you multiclass from Wizard to Cleric, you'd get the spell slots of a 1st-level Cleric, and only be able to cast Cleric spells with them, while you'd need to use spell points for Wizard spellcasting, and only base your number of spell points on your Wizard levels. So essentially, ignore the multiclassing rules for spell slots unless you have more than one divine casting class, and vice-versa for arcane casting classes with spell points.

I wanted to avoid that because it makes multiclassing between divine and arcane casters much weaker and halts the normal spell slot progression intended by the rules, but... I guess it really is the obvious choice

IMO, using Spell Points is one of the best ways (if not the best) to add a little necessary oomph to Sorcerers specifically. Wizards and Bards don’t really need the buff, and it isn’t applicable for Warlocks anyway according to RAW because the Spell Points rules aren’t compatible with Pact Magic. Just something to consider.

Yes, I mentioned that part about warlocks. I also don't think wizards and bards really need the buff, but it's something that fits all arcane spellcasters in my opinion. I'm just trying to figure out what would happen if people at my table tried multiclassing between the two types of casters, wicho would be using different systems by that point

Well, the easiest things would be:

1. Whichever system the starting class uses is what the character uses regardless of multiclassing or not. So a Wiz/Cleric would use points because it was a Wiz first, but a Cleric/Wiz would use slots because it was a Cleric first.
2. Or, whichever the character has more levels in at the time. So a Wiz 3/Cleric 4 would use slots because Cleric has more levels, but a Wiz 4/Cleric 3 would use points because Wiz has more. Ties could get decided by starting class (as above), or by player.
3. Or simply let the players decide which system they preferred when multiclassing and let the use that.