Sorry this is happening to you and your DH OP. 
A few bits of info that might be helpful...
Firstly, he can to go to his GP and get a diagnosis - it could just be a skin tag. GPs can diagnose GWs and also prescribe cream to treat them, although they may as a matter of course recommend that you both go to the GUM clinic to get a full screening. You might also find it hard to get the cream from a normal pharmacist depending on where you live - sometimes it is easier to go to the GUM clinic to get it.
There is currently not test on the NHS for the HPV strain that causes GWs (to the best of my knowledge) so if you don't have any visible warts, you may be a carrier of them and never know.
You can carry the virus and pass it on without ever having any symptoms.
GWs can lie dormant in an infected person for years and years - there is usually an outbreak if the person is run down etc so it is plausible that DH has had it for years.
The HPV strains that can cause cancer are not the same ones that cause GWs and they are screened for now as part of a normal smear - you can get this checked out via a normal smear test but they will only be looking for the strains that cause cancer and NOT GWs.
Is this the only reason you suspect him of being unfaithful?
The reason I am quite knowledgeable on the subject is because this happened to me - I was in your DHs situation - had a GW and have never been unfaithful to DH. Luckily, my DH trusts me completely and had no reason to suspect me of any wrongdoing (there wasn't any) so it isn't really a big deal at all for us - they come and go and when they come I go to my GP and get cream to treat them, when they go we forget all about it. They really are very common.
Good luck with everything - I hope you get to the bottom of it all and there is an innocent explanation.