I've had an email reply from HMRC now and this is what they've said:
Under the provisions of EC Regulations 1408/71, the statement of UK National Insurance contributions was provided on a certificate E104. However, following the introduction of new EU Social Security legislation (EC Regulations 883/2004 &
987/09) from 1 May 2010 we are unable to issue such a statement direct to an individual.
You should contact the foreign institution where you have made your sickness related benefit claim or registration as worker for healthcare and ask that they contact us to request a SED S041 (this has replaced the certificate E104). They should do this by completing SED S040 and sending this request to the following address:
HM Revenue &
Customs
National Insurance Contributions Office
International Caseworker BP1301
Benton Park View
Longbenton
Newcastle upon Tyne
NE98 1ZZ
I trust this has clarified the position.
So now, I need to find out which foreign institution I need to contact. My future employer have said that they don't think the E-104 has been discontinued and are still being as unhelpful. I've now contacted Eures, who are the EU Job mobility portal and once I've got a reply from them, I'm gonna send it to my employer.