Yes, this is correct - they must have your US passport for the entire time, because they need to stick the visa vignette inside the passport when they issue it. So, if they don't have the US passport for the whole time, they can't issue the visa and it's an automatic refusal (they won't request it to be sent, it has to be mailed with all the other documents).
However, it's unlikely to take 3 months based on current timelines - if you are able to pay for priority processing, you could get the visa within a week or so.
Last week, someone had their visa issued just 1 day after the application was received in Sheffield... they mailed their passport on January 26th, the application was acknowledged on January 28th, the visa was issued on January 29th and they received their US passport back on February 1st.
So, at the moment, priority applications are taking 1-2 weeks (or even 1-2 days, as in the case I mentioned above) and non-priority about 1-2 months.