You can apply to the UK Border Agency for permission to marry. They are not concerned whether he is here lawfully or not, just if the relationship is genuine. If you have evidence to prove that you have been living together eg, bank statements, utility bills, etc and photographs of you together then it will help your application. Alternatively, if you are able to marry in an Anglican church then you do not need permission from the UK Border Agency to marry. You mention that your partner is a failed asylum seeker, he may well be under the Home Office legacy case which deals with the back log of failed asylum seekers who are still in the UK. He should see a solicitor about this since the UK Border Agency are making decisions under the legacy and some people are being granted indefinite leave to remain. Either way, you can apply to marry and once you are married, then he will become a family member and protected under European law.
Hope this helps
If he reports his passport lost to the police it is very likely they will check his staus and immigration will be called. He could be detained or required to report at an immigration reporting centre. If you apply to the UKBA for permission to marry, he can use his birth certificate to prove his identity. Once he has made an application then he could report his passport lost in order to get a new one from his Embassy. That way when the police check with immigration it will be recorded that he has made an application although he is likely to have to start reporting.
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