Hi, sorry for the delay. Vaccines and the timing of when we give them, when they can go out and the pups age can become a little complicated so I'll try and make is understandable.
So the DHP (distemper hepatitis and parvovirus) vaccine should be on board and working. This has an onset of immunity 1 week after giving. The second one was given at nearly 12 weeks.
The leptospirosis vaccine is a little different. The L2 datasheet advises the second vaccine is given 2-4 weeks after the first, she was 3 days over this (using a 7 day week). Obviously also as per your question the L4 vaccine covers for 2 more serovars so your vet has essentially restarted her on the L4 - this usually involves 2 vaccines 4 weeks apart.
The question as to can she go out:
With regard to the diseases she can catch off other dogs that we vaccinate against (DHP) she will be, as much as we can tell (good response to vaccine etc) be covered by the vaccines she has had, so interacting with other dogs in this situation for your average dog should be fine.
Leptospirosis is primarily caught from rats and through rat urine - weils disease in humans. It is advised for dogs who are un-vaccinated or haven't received an appropriate course to then stay away from swimming, standing water and anywhere there may be rats. Rat urine gets washed into standing water/drains/water ways hence the risk. As she went several days over I'm not sure we can say she has had an appropriate course.
It may be your vet has advised giving another week as there is a higher risk in your area and they may have seen some lepto cases. I'm in Bristol and Bath and I am relatively happy if a dog has a DHP properly on board and the dates are correct for them to pavement walk and socialise until the lepto course is up to date but this does vary by area.
I would certainly be hesitant to go against your vets advice though as they will know the risks in your specific area.