Not sure if anyone is interested in the last update on this, but just in case... As I mentioned, we were not sure what kind of contact we would receive from the Home Office regarding clearance to marry. I happened to remember the date that the Register's Office said we would be clear to marry and contacted the email address that was provided by the office: MarriageReferralTeamGeneral@homeoffice.gsi.gov.uk
The Register's Office said that they cannot provide status updates so we should contact the Home Office. There was an automated response, similar to the ones for visas essentially saying don't contact us, we'll contact you. A few minutes later I received this email:
"As you have the correct marriage visa the registry office do not need to refer you and your partner to the Home Office. Therefore we will not be investigating and you can proceed with your marriage arrangements."
This is very much what we (and one of the Assistant Registrars) thought would be the case, however, I guess I thought they would notify us (or even the Registry Office) to say 'yay' or 'nay'. Maybe it will be different once the process is in place for a bit longer.