Family Lore Clinic: Are costs claimable in private law proceedings?
The reference in the question to 'private law' suggests that the questioner was referring to children proceedings (but not proceedings involving local authorities, which are 'public law'). Nevertheless, I will deal with all types of non-public law proceedings in this post.
The general rule on costs is that the unsuccessful party will be ordered to pay the costs of the successful party. However, that rule does not apply to family proceedings, where the rule is simply that "the court may at any time make such order as to costs as it thinks just." Further, in financial remedy proceedings the rule is that the court will not make an order requiring one party to pay the costs of another party, unless it considers it appropriate to do so because of the conduct of a party in relation to the proceedings.
So, the answer to the question is that costs can be claimed, although only in limited circumstances in financial remedy proceedings. However, the reality is that family courts very rarely order one party to pay the other party's costs. In fact, in children cases, costs orders are hardly ever made, unless there is an exceptional reason, such as proceedings knowingly being brought despite the fact that they clearly have no prospect of success.
The best advice, therefore, is that you should approach family proceedings on the basis that you will have to pay your own costs.
(As usual, if you require more details or specific advice, you should consult a specialist family lawyer.)