I was going to ask you what the hell this was doing in this thread, but it is, indeed, Jersey talk. And yes, I went through this exact process in May.
I was able to fill out all the relevant paperwork for my car at the DMV without the title present, and then I had to contact my lender and have them send it over. I forget exactly how it went down because there's nothing particularly memorable about standing in a DMV or standing in a DMV parking lot smoking on hold with a bank, but I believe I had to go back because I didn't realize the financing half of it. Happened pretty quick though.
You never touch the title, it's just sent to NJ for the transfer and then sent back to the lender. It really happens as fast as your lender can get it over to the state, so if I recall you'll need to get the info on where to send it from the MVC and then let them know.
Luckily, the inspection is a damn breeze. You just drive up, kill the engine, they basically plug a fancy OBD scanner into your car, turn the key, stare at the screen for 30 seconds, and slap a sticker in your window.