OK, there's a few things to unpick here. I'm an NHS GP.
We do the mother's check and the baby's check at different times. this is deliberate as we find that if we combine them, the Mum finds it difficult to concentrate on her own health, which is just as important as the baby's health.
We do the Mum's check at 6 weeks, earlier if asked, and it covers healing after the birth, mood, is a smear due, contraception and any other concerns she has. For someone who has had severe pre-eclampsia we might see her much earlier to sort out BP meds etc, or it might be that she's still under the care of the hospital and community midwife for that - it varies.
If your hospital team want tests done then they should give you the forms, so that's very cheeky of them ordering the GP to do specific tests. First action is I would contact your consultant's secretary and ask for the forms, so that the results go back to them and not to the GP. We have a massive issue in primary care with secondary care trying to give us unresourced work which we don't have the staff for, plus it introduces extra loops into the pathway which can be dangerous.
We do the baby's check in the week that the baby turns 8 weeks and they get the full check over and their first imms. We are not allowed to give the imms at less than 8 weeks (literally, we'd get into massive trouble for giving at 7 weeks and 6 days) and our baby clinics in my surgery are only one day per week so it would usually be between when your baby is 8 weeks and 8 weeks + 6 days. I suppose it is possible that if something happened at the very last minute e.g. the dr or nurse doing baby clinic (dr and nurse do it together) was ill and we couldn't find cover then we might have to push it back a week- having it at 9 weeks isn't the end of the world.
Not offering the baby check until 12 weeks is unusual. But I would caveat that by saying that primary care is under immense pressure. I personally have several friends who are close to burnout and thinking of leaving. I'm aware of three GP suicides in the last couple of months. I myself would have burnt out and left GP years ago if it wasn't for the fact that I recognised it, cut down my GP hours and do other things for part of the week. So your surgery isn't automatically bad/lazy (they might be - there are good and bad in any walk of life) - on the odds, they are more likely to be struggling with supply and demand issues where not only do the ends not meet, but the ends can't even wave at each other.
We don't sign people as fit back to drive, it's your insurers decision so I'd ring them on that - it's usually when you feel safe to do an emergency stop. Ditto with exercise, I wouldn't say anything more insightful than start slow and listen to your body.
Hope that helps - happy to come back if you've got any further queries, though with the GMC's social media policy in mind I should be clear that this isn't individual medical advice.