I would go and see a solictor regarding the family home and whether she has a genuine legal claim on it. You can get a free 45 minute intitial consultation with most firms.
You have four children aged 8,6,3, &1? I think it's only natural to want to see your kids, but you shouldn't do anything that your ex may be able to use against you. I know it's horrible, but this is how it has to be for now. You certainly don't have to apologise for having been angry, upset or depressed about losing contact with your family. It's a truly terrible experience.
If you can't afford a solicitor or don't qualify for legal aid, you can represent yourself regarding contact and you'll find a comprehensive guide at the top of the Legal Eagle forum that one of the other Dads produced.
If the mother is not entertaining mediation you should apply for a contact or residency order, which will have to be filed at a court nearest to your ex. The residency order would be more difficult to obtain, because the courts are unlikely to remove your children from their mother unless there is a genuine welfare risk. However if you were to look at a shared residency arrangement you may have more success. This is essentially a court order that recognises that although your children live with the mother for the majority of time, they also have a second home with you.
The mother has put a significant distance between you and your children, so you need to think about realistic and practical forms of contact. If you want daily contact with your children this is more likely to happen via telephone or Skype. You can expect to get regular overnight contact at weekends, usually alternate weekends. However you're looking at a 600 mile roundtrip and you'd need to think about how this can happen regularly. Will she agree to meet you half way? You want to make sure that you can have extended weekend contact over bank holidays or other special occasions. You also need to think about holidays and how things like birthdays and christmas are going to work. I know it's horrible to have to think in these terms, but this is how things are going to be in the immediate future.
It may be that you consider relocating yourself, so that you can be nearer. However I would wait and see how established the mother is in her new area. You don't want to relocate only to discover that she's going to move again.
I think this is a cruel situation to be in Dave and I'm truly sorry that you're in this position. It's horrible.
You're not on your own though and you'll always find some support here 🙂
Best wishes,
FM '70