In my experience so far, the lease company gave Tesla my contact details & Tesla have been sending me messages directly regarding RN, VIN, Reg No. & delivery date, etc. Check with your lease company if they used a referral code to order your car - if they can use one, you (and the referral code owner) will get 1,000 free Supercharger miles in your Tesla Account when you get the car.
I had an order confirmaton text with the RN on 23rd June which also advised that a Tesla Account had been set up. You won't see any details for your allocated car in your account until after you have signed for the car on delivery/collection. I received nothing from Tesla for several weeks until the middle of August with a text confirming a match, with a delivery date & asking me to confirm my preferred delivery address. Last Monday, I received a text with confirmation of VIN, Reg No. & delivery date (9th Sept). I don't think you get this info until you confirm your delivery address or accept a collection date/location.
It's worth noting that if you don't reply to the text asking for delivery address, Tesla will keep sending reminders & will eventually advise that if you don't confirm in good time, they will cancel your delivery date & allocate you to another car for a later delivery/collection. Also, it seems that all leased cars are delivered where possible.
Hope this helps & that you get something positive soon!