Does he intend to move back to the US?
He needs to stay in the UK.
Where is he from in the US?
Does he have positive feelings/ attitudes toward guns?
Any survival / wilderness experience?
Military training?
He absolutely MUST let the children call you, and you MUST be allowed to call them (within reason - not nine calls a day, monopolizing them for hours, etc).
This is really important. Make sure the judge is made aware of the fact that he's playing keepaway here, exhibiting signs of possessiveness and jealousy and vindictiveness toward you by refusing them permission to have contact with you.
File a motion requesting he issues an order requiring that you both allow contact by phone, text, email, and using features like Zoom, FaceTime, WhatsApp, etc, during each other's parenting time. He must be allowed to contact them, and they must be allowed to contact him while they're with you, too, in other words.
I think 3:3 is a fair division of the summer holidays, especially since you're currently doing 50:50. A three week spell in the US would be fair, I think, with two days of travel time included in the counting. But he would need to bring them back. If there's anything in your past relationship that indicates he would bolt and refuse to hand them back, you need to make it clear that this is a possibility.
You will want to revisit this if the DCs get summer jobs as teens.
Make sure the supreme importance of not interrupting the DCs' social or school or employment life or extracurricular activities is emphasised. You and he will need to cooperate when scheduling so you don't end up being signed up to schlep them to Cornwall for a week of surfing in May, for example.
When I divorced (albeit in the US), we divided up the year into ordinary time and holiday time.
Basically, he got EOW.
We each got every second Easter.
He got Thanksgiving and the day before Christmas every year, and I got Christmas Eve and Christmas Day every year.
He got twelve days during the summer holidays (two full work weeks plus the weekend in between). This was twelve days out of the 12 weeks we get here for summer.
The DCs got two weeks off school for Christmas. I got the first half of the break, and he got the second half, with the Christmas Eve and Christmas Day thing interrupting the first half usually.
All birthdays were celebrated on the day of the birthday at whatever home the children were in on the day. If a birthday party was organised, with invited guests, both parents had to have the option to be present and their schedules taken into account when organising.
I had them for Mother's Day, and he had them for Father's Day.
Visits by grandparents or other relatives did not interrupt the schedule.
Permission to travel would not be unreasonably withheld for extended family events, specifically weddings or funerals. Not destination weddings, however. 50th anniversaries or 80th birthday parties were by negotiation, taking into account school attendance regs, any social or extracurricular activities the DCs had. There was a hierarchy of extended family events, in other words.
The holiday time was in brackets and the ordinary time pattern of EOW was uninterrupted - for example, I had them the weekend before Easter/ Thanksgiving, he had them for Easter/ Thanksgiving, then he also had them the weekend afterwards. The aim of this was that nobody ever had them three weekends in a row including a holiday weekend.