The issue will be that you are on the second leg of the trip, and do not have a ticket showing in the system to leave the UK. Whilst you are allowed to enter the UK without a visa, there is a time limit to that on a US passport (without a work permit) If a roundtrip is started in the US it will not be an issue. If the roundtrip was started in the UK, the system will flag it - until someone overrides it (either by showing a valid work permit, a valid second passport entitling you to remain in the UK, or an onward ticket showing you will leave the UK within the allowed time)