After 6 weeks in the hospital, mom finally transferring to private nursing home. Should I be there on arrival?

I'm assuming she's going into LTC...I would say yes to hang out there and just make sure everything gets transferred correctly and she gets as settled as possible. She won't be happy for you to leave, but don't stay the night if you are not concerned about her health or med schedule, etc. You can play it by ear the following day(s) but a lot depends on if the facility is getting her medical needs and prescriptions correct and it's important to make sure they are. I also think it's good to introduce yourself the the staff and get to know them a little so you know who to ask for if there are concerns. I wish you all the best as you help your mom through this transition!