I has a similar experience - 3 letters to SS in Baltimore without reply - disputing my earnings records from 1993!
One conversation with FBU who then guided me on what form to fill out and records were corrected within days - worth an $45 per month on my (maximum WEP'd) SS retirement pension for the rest of my life
FYI, when I spoke to the FBU recently they were adamant that I should always send correspondence to them (and that's especially true when sending originals of supporting documentation) rather than to the USA. They have the ability to upload copies of that correspondence to the SSA. I wasn't aware of this before.
The "signed for" letter I sent to the FBU is still not showing anything other than "we've got it" on the tracker, which leaves me in a bit of a quandary. Either the letter has not been delivered (3 days later) or it's been delivered and not been signed for - neither outcome is acceptable. It may be that no letters to the embassy can be signed for (I can imagine a mailbag arriving and no postie is going to hang around waiting for letters to be signed for - even if he/she is allowed to enter the embassy), so it may very well be that it's not possible. If that's the case, Royal Mail shouldn't offer such a service when sending to the embassy. Suffice to say, I'm not a big fan of Royal Mail - I've had loads of issues over the years.
Update: received the "delivered" notification today, so that's 4 working days. Considering it was within the UK, that's pretty slow but delivery could be complicated at the embassy end. At least it's arrived.