For children born pre-December 2003, unmarried father's effectively have two choices for obtaining PR.
The first is to enter into a PR agreement with the mother. It follows, as a matter of logic, that the mother must therefore consent. A formal document is drawn up & registered. Simples. As such, the cost would be quite low and they would probably not have to see each other.
The position is slightly more complex if the mother doesn't consent. As has been said above, an application to the court would be required. This is not difficult, but would likely require attendance at court on at least one occasion, by both parties. Again, as has been said above, he might be able to use the first court hearing for conciliation, but the back ground of the case & the willingness of each party to mediate will determine this.
How long the proceedings take will dictate the cost. If they are contested all the way to final hearing, then the cost is likely to increase. A decent solicitor costs anywhere between £140-£200 per hour, possibly more, depending on your location. He may be eligible for legal aid, depending on your income.
When deciding whether or not to grant PR the court always considers the children's welfare. The reality is that if a father has shown commitment to a child, he is likely to be granted PR. If your partner has fairly regular contact with them and plays an active role in their lives, I imagine a court would need to be persuaded by the mother that there were excellent reasons to withhold PR.