BA Executive Club Silver corresponds to Oneworld Sapphire (OWS) which enables lounge access to Business Class lounges regardless of your class of travel. It also entitles you to one guest. (You won't get access to First Class lounges where they exist)

So, the short answer is yes, you'll have access.

You can look here to find the specifics at each airport: Airport Lounges: oneworld Lounges & Airport Lounge Access | oneworld

At Heathrow Terminal 3, you will have a choice of lounges (tho I believe AA is still closed, and few people would find any reason to recommend BA over Qantas or Cathay)

EDIT: Beaten by a minute But Never Stanstead has given you the info you need. You will generally find any exemptions buried in the small print of the Oneworld Lounges web pages that we have both linked to, so have a look at the details in there