Not yet but my wife's phone (whose number we stupidly gave to them) has service issues and text issues so we don't know if anything has been received. The DX lady at the embassy was confident when she said two weeks
It does depend on what type of passport you're applying for as to what the processing times are.
Current processing times on the London Embassy website (which I understand are pretty accurate):
Report of Birth Abroad and First Passport:
- Processing time for a Consular Report of Birth Abroad and First Passport is approximately 4 weeks.
Born outside the US and applying for passport:
- Processing time for a full validity (10 years for adults; 5 years for children under age 16) U.S. passport is approximately 4 weeks.
Renewing a child's passport:
- Processing time for a full validity (5 years for children under age 16) U.S. passport is approximately 3 weeks.
Renewing an adult's passport:
- Processing time for a full validity (10 year for adult) U.S. passport is approximately 3 weeks.