Here in the US they start at 6 weeks and do injections at 6,8, 12, 16 weeks. You MUST NOT be late on their initial shots after they receive the first one because they will have to start all over again. With the situation of the mom, the puppy needs to be checked anyway. Hopefully, the pups teeth were getting too sharp for the mom and she was weening them because they were biting her, maybe that's why she was trying to get away from them. I can't say I blame her, I'd run too with all them sharp teeth biting me LOL. Enjoy your new Puppy.
All states and countries are different, i do in california some vaccination can be given at 6 weeks and some you have to wait until they are 8 weeks most for the time they wait until 8 weeks old.What I would do is talk to your vet.The vet can give the little girl a good check up and make sure she health and also put you on a vaccination schedule and tell you when to start with the shots the deworming ect.So in till then keep her inside, do not take her to parks, do not let to may people especially with dogs hold her, keep her away for dogs and take her to the vet.