If it were me, and you have to be at the hospital at 10.30 am, I’d book a room in a Travelodge or Premier Inn, near to the hospital for the night before. Then, it’s a nice cooked breakfast and an easy 10 - 20 minute drive to the hospital. Years ago, I stayed at a Premier Inn on the A52, going towards Derby - iirc, it was a 10 minute drive to the University, which is near Queen’s Hospital.
Travelling on motorways is ok, when nothing goes wrong. We travel regularly on the M1/M6 or M1 and come off just before Derby, go along the A50/A500 to the M6, to avoid going round Birmingham. It should take 3 hours and a bit. It can often take 5 - 8 hours, depending on how many crashes there are; and especially if they close the motorway.
I can’t recommend any service stations up the M1 from the M25. It’s just a personal preference which fast food outlet, you like best. Watford Gap has McDonalds, Costa, Chozen Noodle and Leon. Leicester Forest East has Chopstix, Taco Bell, Burger King and KFC to name a few. Rugby is supposed to be the best, but it’s on the M6. We use Google maps to look for cafes and restaurants, just off the M1. It’s usually better quality food, and better value for money imo. Alternatively, take your own food. Flasks and a croissant are certainly cheaper than Costa or Starbucks, the main offerings at service stations!
Take a map book too. When the motorway is closed, it’s very annoying if the satnav keeps telling you to go back to the motorway - especially if you are in a part of the country you don’t know at all, and don’t know where to go!