The GP is the place to start. Perhaps she could see a different GP at the practice, initially? Her DH will need to be seen eventually, but most GP's are willing to get the ball rolling if the woman attends alone.
Eventually her DH would have to be seen to rule out male factor infertility, but as he has children already then that does seem somewhat unlikely.
These are the initial tests that my GP ran:
Rubella antibody test (to check I was immune before becoming pregnant)
TSH (thyroid stimulating hormone) to rule out hypo or hyper thyroidism that can be a cause of infertility and can be treated by a GP initially or a specialist endocrinologist
Fasting blood glucose (to rule out diabetes)
Bodymass index (being very under or overweight can affect fertility)
Follicle Stimulating hormone (FSH) and luteinising hormone (LH) levels - high FSH can be indicative of menopause or ovarian failure, high LH can be indicative of polycystic ovaries
GP should check her periods are regular (it's a good indicator of ovulation)
Day 21 progesterone (blood test done 7 days before your period is due to confirm that you are ovulatory
STI check - they are looking for chronic chlamydia infection that can be indicative of tubal damage
The GP will also advise her to take folic acid.
Following initial tests, which I am pretty sure she can demand from her GP, the next step would be a sperm test for her DH, and then a referral to the fertility clinic where they will arrange an HSG for her.
Where I live the fertility clinic absolutely insists on seeing the couple together at this stage as the medical history of both is important.
Hope that helps.