Keep things the way they are until there is a resolution and it is something you can live with.  Do not sell your house or put his name on it or move in together.  Definitely do not get married or pregnant.  Physical custody, which could take a while if she fights it even with a free attorney, is not certain to be awarded to him.  At their ages the kids will have a voice in where they live and the oldest may even age out during the proceedings.  See how he handles this and whether he tries to baby or buy them or if he treats them in a reasonable parent/child manner.  Keep your eyes open and get a good look at your future.  You have been given a gift here.  Make sure you take it.

Welcome to the site!  Considering the ages of your bf's kids - there may not be even any jurisdiction by the courts - there probably wouldn't be in the UK where I live - but I know the USA is a bit more inclined to court orders generally than this country is.    I would urge you to keep your own house and don't move in with bf until things are clearer.  Even then - I would not get rid of your house - if you own it, I would consider renting it out. 

Most people on StepTalk would probably advise you strongly to leave yourself a clear escape route - as step set ups can become very challenging even if they don't seem so to begin with. 

Welcome to Steptalk.  I have been in your shoes. My DH won full physical custody of his kids in 2015.  It was a huge adjustment to our family. We already lived together and were married. It almost destroyed our relationship.  The reason being that BM was so angry over losing control of skids and losing her revenue stream (CS) that she pinned the skids against us.  It can be a major adjustment so I would say that the other posters are correct. I wouldnt move in with him or get married until this mess is settled. It could be a long and expensive process. Also, you have no idea how BM will react.  The Skids may also act out if they feel that their mother is being pushed out.  Once this is settled, you can find a way to combine your lives. Trust me, I wish I would have been smart about this when my DH wanted to go for custody. It was the worst period in my marriage.

Where I live, getting custody changed is never a slam dunk.  The courts are supposed to look at the best interests of the child, but that does not always seem to be the case.  Your DH may end up finding this is time and money consuming battle. 

In any event, you and future husband need to be open and frank of what financial, emotional, and other responsibilities you each have.   When I got engaged (my first DH died), I explaned that my late husband's  insurance would provide for my DD to go to college.  Current DH had no college savings for his DCs.  That is a very difficult situation.    I agree with PP, take your time, talk things out first.   I had expectations for my DD with helping around the house.  Make certain you guys are on the same page.  His kids may be somewhat damaged by the less than competent parenting from mom.  If that is the case, I would recommend therapy, but, IMHO, thereapy only works when people want it to work. 

You are only two years from the initial launch point into empty nester status.  Discretion being the better part of valor, you may want to forego cohabitating until the youngest turns 18.  That eliminates any legal need for the Skids to be cared for or resident in your relationship home.

That said, if you do choose to move in, set the rules of reasonable behavior, enforce the rules and you and SO be united as equity life partners and as equity parents to any kids in the home regardless of kid biology.

Good luck.

Your SO is foolish to start a battle for skids who are so close to aging out. Even a "slam dunk" with lots of documentation will cost several thousand dollars at minimum. Steer clear of this mess.