No skill proficiencies gained from multiclass into Artificer
The 2019 UA Artificer includes, at the very end, a sidebar on how multiclassing into artificer works if you're using the optional multiclassing rules:
Optional Rule: Multiclassing
If your group uses the optional rule on multiclassing in the Player’s Handbook (p. 163), here’s what you need to know if you choose artificer as one of your classes.
Ability Score Minimum. As a multiclass character, you must have at least an Intelligence score of 13 to take a level in this class.
Proficiencies Gained. If artificer isn’t your initial class, here are the proficiencies you gain when you take your first level as an artificer: light armor, medium armor, shields, thieves’ tools, tinker’s tools.
Extra Attack. The Arcane Armament feature doesn’t give you an additional attack if you also have the Extra Attack feature.
Spell Slots. Add half your levels (rounded up) in the artificer class to the appropriate levels from other classes to determine your available spell slots.
These are also reproduced on the DnDBeyond class page for the artificer. Unfortunately for your multiclass, you don't appear to gain any extra skill proficiencies by going into artificer; only the tool proficiences for thieves' and tinker's tools.
The original UA Artificer from 2017 doesn't include multiclassing rules (as NautArch notes in comments, UA classes normally don't) so if you're still working from that version, that might be why you can't find these rules.