Hi dragonfairy, so sorry to hear what you're going through.
You can file for divorce in the UK on one of the following grounds:
1. separated for 2 years, divorce by mutual consent;
2. Adultery;
3. Unreasonable behaviour;
4. Desertion (for minimum of 2 years);
4. Separated for 5 years, divorce without consent of one party.
The papers can be served on him in Indiana or where ever else he ends up.
As for custody, that can become complicated where one parent lives overseas but given that he has already walked out and left the country I'd say his chances of getting joint custody are not good.
Do the kids have US passports? If so, who has them, you or your husband? If he has them, you can obtain a court order to force him to surrender the passports.
I don't know what the situation might be in Indiana, however, as others have said he may be able to obtaion a divorce there without your consent. As regards custody, again, I've no idea how Indiana would treat this situation but my gut instinct is that since he has left the children behind in Englan he'd be out of luck there.
However, as the others have said you are going to need to get specialised legal advice on this. If finances are an issue, contact CAB who can give you a list of legal aid lawyers in your area. Please feel free to PM me if you have any more questions.