You gain no skill proficiencies by multiclassing into Artificer
Per the Artificer class description on page 54 of Eberron: Rising from the Last War (or in Appendix D of the Wayfinder's Guide to Eberron), there is an optional rule addendum on how multiclassing into artificer works if you're using the 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, or to take a level in another class if you are already an artificer.
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.
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. Multiclassing rules were first added in the early 2019 UA version (as a sidebar at the end of the document) and finalised with the release of E:RftLW.