The mother still has parental responsibility. That means you need her consent before taking or sending your step daughter out of the UK. If she refuses to consent your husband can apply to the courts for a Specific Issue Order to resolve the matter, raising the mother's refusal to hand over the passport. The order would almost certainly be granted.
Having been in this situation recently, I was told by the UK Passport Office that if one parent was refusing to hand over the passport, on an application stating the circumstances they would issue a new passport as the old one was unavailable. There's probably a number of factors differentaiting this from the advice given by the other posters, but thought that I'd throw this in as it may be a quicker way out. Suspect that it differs if an order is in place or not.
Though if you're holding on to the passport for a reason (eg other parent threatening to remove the child without consent) I know from the experience of a client of mine that they'll put a block on the issuing of a new passport - in this instance because someone with parental responsibility had possession of it so it wasn't lost.
Only the parent who originally applies can get a replacement (ex applied for my ds and I looked into it) which means she can say it is lost at the last minute and it's tough. My solicitor suggested writing to ex asking politely for the passport to be given by a certain date. Try this first before getting solicitor involved.
I don't suppose she will turn 16 between now and the August trip will she? I wouldn't advise hanging around for letters to be sent and a possible reply, if she has unequivocally stated she is not returning the passport for the purposes of the trip you do need to make a court application ASAP (on a C100) to get heard before August. As baby barrister has said it will almost certainly be granted given your DSD's age and the circumstances of the proposed trip. It is possible you could cancel and obtain a new passport, but that does not get round the problem that your DSD won't have her mother's consent for the trip and really she does require the consent of all persons with parental responsibility. Although as it is not your DH taking her out of the jurisdiction, if she were to get her own passport and go away he hasn't committed the criminal offence of abduction although there might be some aiding and abetting type offences. Unlikely it would be pursued but really the best thing to do would be to make a court application and get the passport back properly. You could easily do it yourself without a solicitor - it is a straightforward application/issue.