FWIW, the UK and Ireland for many years have a common boarder. This means all flights between the ROI and the UK are considered domestic. Technically this also means a passport is not required for travel between the two, however in practice most people bring their passports, especially as when you fly into DUB from LHR you still pass the boarder checks. Coming back into LHR however there is nothing!
Most EU countries are now part of a boarder agreement called
Schengen which effectively means that flights between Schengen countries are effectively considered domestic and therefore do not have any boarder controls. The UK and the ROI of not part of this agreement therefore a passport is required to travel between the UK/ROI to Schengen countries such as France and Germany.
To answer the OPs original question, if your flying through T1 with BA and have OW status, you have access to Zone R which provides a Fast Track service regardless of class of travel. In T4 however, BA only provides access to Fast Track for passengers flying in F & J.