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The domestic connection doesn't matter in this context, while in T3 you remain outside the UK Border. You can either clear the UK Border when you finally get to T5 via the airside bus, and then through the First Wing (which is best if you can use the e-passport gates0; or use the UK + Ireland Flight Connection route at T5 instead; or exit T3 landside, use HEX and then re-enter via the First Wing in T5. Welcome to the BA forum.