I'm afraid I can't help too much on the practicalities of your problem but I can make a few points.
I assume your aim is to discover if your partner has another child in order to release the obvious tension this might cause (if it's simply to avoid maintenance then this has already been done - as the mother is refusing tests (on behalf of the child) it is presumed that your partner is not the father).
In order to do this, you can go to court seeking an order forcing the mother to allow DNA tests. In recent years the courts have become very receptive to such applications and will make the order wherever it would ease any stress in the parties; it is also thought in the vast majority of cases that a child should know whom his/her parents are.
If the mother breaches the order, she will be in contempt of court. Unfortunately the sanction in most cases, as you said, is a fine or prison rather than direct enforcement (i.e. enforced testing).
The best thing is to visit a solicitor and ask for a detailed breakdown of the costs involved (solicitor fees, barrister fees, court fees etc.). Also consider the effects on you and your partner carefully - discovering you have a child will create emotional demands, leave you liable to pay maintenance etc. I would tread carefully before choosing to seek an order.