Definitely take the cautious approach. My friends recently arrived into LHR and witnessed a couple who got it wrong, kicked off upon discovering that they will be put into hotel quarantine, and had to be escorted away by security.

Based on your dates, and assuming you actually arrive in the US on 2 October, then you can arrive in the UK after midnight on 13 October onwards, as that gives you 10 clear days in the US. However, if you flight to the US is overnight and arrives on 3 October, or if your flight to the UK is a day flight and arrives on 12 October, then you no longer have 10 clear days prior to the day of arrival in the UK.