1. Your question is unclear as to what you want to ask. There are no legal costs payable by beneficiaries under either a will or under an intestacy (no will) situation. Once these two daughters are shown to be his children, such as by production of their Birth Certificate, then they inherit. Be aware that the partner has a right to apply under the 2011 Civil Partnership & Rights of Cohabitants to get the equivalent of a legal right share of her former partner's will. In this way, the division of the estate roughly equates with that of married couples and their children. If you want to provide any more detail, then I will clarify any further points you raise.
2. Was there a will?
3. If there was no will, then it is akin to a married man dying. The wife gets two thirds and the two children one-sixth each. The wife would have to go to court to get her two thirds share in the absence of agreement with the person who extracts letters of administration to the estate. However, as the partner will get her legal costs paid from the estate of the deceased, forcing the partner to go to court will only deplete the amount of money for everyone. So I would recommend agreement. You can consider taking out letters of administration to your son's estate and set the thing in motion.
4. Then the best course would be for you to extract letters of administration so that the estate gets admitted to probate. Then seek the partner's agreement that the estate will be split as if she were married with her getting two thirds. I would suggest you let the partner appropriate the house for her share with the money going to the kids and the pension shared. That would be treating the partner akin to a married woman.
5. The costs to the children will come out of the estate of your deceased son. No one stated this because this is the way all the lawyers get paid. You need to consider whether paying part of the estate to the lawyers justifys seeking to assert the children's rights in this fashion.