Service Records Directions
UK and USA
UK Forces Research Links and military service records
The primary source of military records for the UK is the Ministry of Defence at the link shown below. You can print The form off (I don’t think you can apply online) and send it off. They warn you on the site that it takes a few months but in practice it’s more like a few weeks. It’s £30 for the enquiry but if you are next of kin then it’s usually free. They ask you for as much information as you’ve got, name, date of birth et cetera, maybe regiment and service number if you have them, though I’m pretty sure that’s not compulsory.
Ministry of Defence at https://www.gov.uk/get-copy-military-service-records or ring Veterans UK at 0808 1914 218.
You can get more information from Veterans UK at https://www.gov.uk/government/organisations/veterans-uk.
You can also get hold of the unit war diaries which detail the daily activities of the battalions.
Other link which may offer useful background:
A secondary source of information might be the regimental association or any museum associated with the regiment in question, if known. For instance, the Green Howards Museum in Richmond, Yorkshire can be a useful source of enquiries about the Green Howards.
US Forces Research Links and military service records
Please find below a few links which might help your research. Not sure what US archives exist but WWII Forums is a great forum to research and ask questions. http://ww2f.com. Sorry but this is all I could find but Google might uncover other stuff.
Essentially, you'll need to track down the service records and I'd try here: https://www.usa.gov/military-records
I guess they'll need full name and dob as a minimum to assist them.
From there, if you succeed, your research options will expand massively.