Hi Nades, have you mentioned this to your plunket nurse? She has the power to refer you on to a paediatrition or a OT/physio if she thinks it is an issue. Or go back to your doctor and push your case. I know some kids hate tummy time, but all the ones I know that do have still been able to reach milestones at the normal times, and lift their heads by about 4mo.
We have just been through this from about 3months with our wee girl, she turned out to have a torticolis (google it for explaination!) and also a muscular problem. Since then we have been seeing the paediatrition and the neurodevelopment therapist (like an physio for children that are developing slowly physically) and she has improved in leaps and bounds, although she will always be a bit slower with gross motor skills. Once she eventually started holding her head up (about 4-5 months with exercises to strengthen) she improved rapidly, and once she started eating (we did BLW from 5.5 months) her fine motor skills improved heaps. Now she is nearly 14 months old, her gross motor skills are that of a 9-12 month old, but her fine motor skills and cognitive skills are that of a 16mo.
I hope that your girl is just taking her time, but I would definately get everything checked just in case. Sorry, hope this doesn't scare you - but its better to get onto anything earlier rather than later though - and it may be nothing at all anyway