A US national who is already in the UK does not need a visa to give notice or to marry in the UK. This assumes the partner is a British national or settled in the UK.
What does affect you and your fiancé is the change that came into effect in the Immigration Act 2014, and as explained in a Ministerial Statement published on 24 Nov 2014, which says. in part:
All proposed marriages and civil partnerships involving a non-EEA national with limited or no immigration status in the UK, or who does not provide specified evidence that they are exempt from the scheme, will be referred to the Home Office. Where we have reasonable grounds to suspect a sham, we will be able to extend the notice period in these referred cases to 70 days in order to investigate and take appropriate enforcement or casework action where we establish a sham. A couple will be unable to get married or enter into a civil partnership on the basis of that notice if they do not comply with an investigation under the scheme.
It means your notice period will be 30 days, with a possible extension of up to 70 days. Not every case is delayed.
You stated that:
I can't find anything that supports what we have been told by the lawyer - does anyone have any information on this?
Your lawyer is correct. You can read the act linked above or review Parliament's debates on the subject. A comprehensive briefing is available to the UK legal community (behind a paywall) on the topic, which is quite probably the reference cited by your lawyer.
Once your marriage is celebrated, your new spouse will be able to apply for a settlement track in the FLR (FP) route. If successful, your spouse will be on the 10-year route to permanent residence (rather than the 5-year route).