Long post, but there's lots of practical advice! Been there, done it, paid the 6 figure legal bill.
I'm so sorry for your situation, OP, it sounds like hell.
But you really don't have to let him dictate the pace.
You definitely need to see a solicitor, sooner rather than later. To prepare for that, get your ducks in a row - all the financial paperwork, all the info about who does what in the house but particularly with regard to looking after the children.
Also give thought to what you ideally would like as a financial separation and regards custody of the children.
And what you are prepared to accept/negotiate.
Definitely go for a 'clean break' ie no maintenance payable to either party.
As you earn more than him, and probably have a good pension, he might go after you for maintenance and a pension share, for example. Try to anticipate what he might ask for and prepare your answers.
For example, you won't ask for any CMS even if they're with you most of the time (if that's affordable for you), but in return you wont be giving him any maintenance or pension share.
Family court like couples to try mediation first, but they do not like it when one party drags their feet or doesn't fully engage. Email him again to ask when his appointment is.
Make sure you put any requests to him in writing- text or email. Screenshot everything and print it out if you can.
Email him with attached copies of the 5 estate agents valuations. (I've been involved in 2 separations and both times only 3 valuations were required by the Court. The average of the 3 or 5 or however many is taken as the price). Give him a deadline to organise other viewings.
Also, prepare examples of other houses locally, what they have sold for recently or are listed for (EA may have included this in their valuations already).
Did the EAs say the unfinished bathroom was having a big impact on the house price? If so ...
Email him re the bathroom. Tell him you understand he would like to fit this himself, which is about a two day job (unless tiling required also). However he has done nothing on it for the past two years. Give him a deadline to complete the work or you will find a builder to complete it.
Aim to be ready to file for divorce immediately after mediation has happened (if it ever does). The person filing has more control over the process, so it's worth doing. Also do not give him/his solicitor pre warning of it, just do it.
He keeps reminding you to keep it "amicable" for the children's sake. This is translating into you basically doing everything for them, facilitating him living the single life, and accepting him delaying the actual separation/divorce.
Courts don't really care if one party is already involved with someone else but they do care about the impact on the children's care. So keep a record of everything you do. Every time you ask him to do a drop/pick up and he says he can't - get it in writing (if you haven't already). Keep recording everything. That is powerful evidence in Court. And this WILL end up in Court, I'm sure, because he is not co-operating already!
The Court's watch word is 'reasonable', and their priority is to ensure the children are provided for as best as possible.
Lastly, see your GP. Tell them how upsetting/stressful this situation is for you (and the children, if it is. It surely is adversely affecting them).
Your DH is living a single life, seeing another woman, almost no help with the kids, dragging his heels re everything etc.
Ask if there's any help they can give eg medication, counselling...
You could then possibly apply through your solicitor to the Court for an Ouster order ie making him move out, using this as evidence that the current situation is untenable for you and the children.
This isn't going to be easy, OP, but if you focus on what YOU can do, and start moving forward with your own plans you will start to regain control of the situation and the process.
Good luck! 💐