I went through this when I bought my Model Y in October (OR pick up, WA resident). Ultimately it took about 45 days or so for me to get it, and that was me calling the Portland store, the state of Washington, a Washington store, and the title and license office the WA store uses. There’s no reason this takes so long. I had my reg and plates within about 3 days of the WA registration office getting the paperwork and payment from the Tesla WA store. The slow down is on Tesla. I forget which location they send the stuff to, but it gets sent to a WA Seattle store. They process it in batches and then take all the paperwork and cut a big check to a local registration office in Seattle once or twice a week and the office will process the paperwork (in my experience within a day or two) and then send all the plates to the WA Tesla store which then mails out the ones they need to.
I didn’t trust the covid exemption because I was told that was only for when license and registration offices and DMVs were closed. Many (not all at the time) had opened again. What I did was a purchased WA temporary travel tags. They are 3 day temp tags meant for purchasing a vehicle in WA and driving it out of state. You can buy 3 at a time (total of 9 days) and they’re $33/each. You can not use more than 3 of them on the same vehicle in a 30 day period… but they sell them to you without asking any details on the vehicle, just checking your driver’s license. I went to two different offices and bought six total (~$200) and then used two each week (put one on Mon and it’s good till end of day Wed, put next one on Thursday and good end of day Sat, I didn’t drive the car on Sunday, repeat).
That trick got me the 3 extra weeks I needed after my 28 day Oregon temp permit expired and thankfully my WA plates showed up by then.
OR will not issue another temp tag and they actually make you sign an understanding when you’re a WA resident and pick up the vehicle in OR.
WA will not issue you a temp permit unless YOU pay the taxes, they don’t trust that Tesla did even if you have proof Tesla collected them. I was advised not to pay again by the state of WA as refunds were taking 6 to 9 months.
I would recommend against picking up in OR unless you have a second car that can be primary for another two or three months OR you work from home and can limp along driving the car just two or three times a week. Then you probably can squeeze another three to six weeks out of it by the WA Travel Tag trick (kinda sorta illegal the way I did it, legal if you can make 3 tags stretch 30 days before needing more.)
Edit: University Licensing Agency was the location that processed Tesla registration as of Nov 2020.
Licensing Agency - Seattle, WA - University License Agency
You could try calling them and seeing if they have you paperwork or see you in their system and just haven’t had payment submitted yet. This also probably means University Village Tesla is the one that handles OR sold cars since I remember thinking the locations were close by. They are extremely cagey in my experience with actual connecting you to the lady (at the time) that processed the paperwork. Most calls was just some clueless sales rep that would put me on hold and “text her” since she was working remote a lot and then get back on the phone to say there was no reply, or she had said everything was in order and nothing to worry about. No one could understand that I wanted it done quickly since my tags expired. ONE day I got lucky and someone was busy but said they could transfer me to her, I left a voicemail and she seemed confused when she called me back (like not used to customers actually reaching her) and said she found my paperwork and everything was there and not to worry… though she still took over 7 days before submitting it to the licensing office from my daily calls to them…