Not sure whether the pains you describe are down the back of your legs or the front.
If you were only 3 months after birth, your ligaments were probably still loose and susceptible to damage especially if you were still BFing.
You may have pulled your SI (sacro illiac - hip joints at the back) joints. As the some of the pain you are describing sounds like the pain you can get from pelvic instability during pregnancy and after birth.
The pain you describe in terms of walking, laying and turning in bed etc, sounds exactly like the pain of pelvic instability in the SI joints which I suffered after having DD.
You need physio. It will help and you will get back to normal.
You probably need to search around and find a physio which specialises in post natal physio.
Don't expect the NHS to be helpful. I was in terrible pain for months and my GP just kept telling me it was normal after birth even though I couldn't walk up stairs - I had to crawl up on all fours (so handy with a newborn and a house on 3 levels).
When I eventually pushed and pushed them (for that read: kept weeping in their surgery). I was offered crutches and very strong pain relief and told I could apply for a Blue Parking Badge in 6 months time. No other help was offered - useless!!!
I went privately to a physio recommended to me and it worked wonders. I needed about 10 sessions on a 1:1 basis first with a Manual Physio (does massage but its more than that IYSWIM) and then with a Rehab Physio to continue to get my pelvic area to move and function normally again.
I continued after that to do Clinical Pilates session (not the sort you do in a gym but the sort offered by physios) at £25 per class for a while. Plus I have lots of sheet with exercises I can do at home if I feel any twinges or need maintenance.
Until I saw the physio I felt so down and literally thought I was going to end up in a wheelchair (thanks Mr GP!) and I struggled to enjoy the early months with my daughter. The physio changed everything. I have had no pain for many many months now and can jump about as I choose as well as now being 17 weeks pregnant again and lugging a heavy 18 months old DD around.
Honestly, you will be fine but do sort your physio out soonest. Its not worth waiting around in pain.
BTW the exam your GP did with you lying down and they move your legs around is an outdated useless way of looking to see if you have SPD. The reason its pretty poor is that it will only pick up an issue if your pelvic instability in cause by the pelvic joint at the front of your pelvis (directly above your erm flower IYSWIM). It will not pick up issues with SI joints at all. Most GPs do not have a clue how to diagnose this.
Hope you're able to find decent physio in your area soon.
Wishing you the best.