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The name SPEAKER dates back to the 17th century, but there has been no destroyer with the name.
However, I think the answer is provided by T P Stopford in his book 'Admiralty Ships Badges'. He states that the carrier SPEAKER took over the badge formerly used by the destroyer 'Whitley'.
This was a curiosity - the destroyer, built in 1918 was supposed to have been named 'Whitby', but thanks to a typing error, she was launched as 'Whitley'. It was decided to keep the name, although it was unclear what or who she was named after. In 1921 when the badge was being drawn by Charles ffoulkes, he realised that the newly elected Speaker of the House of Commons happened to be Mr Whitley - so the mace was used as a badge.
The destroyer was sunk in 1940.
In 1943 when the carrier was built, she took over the badge (presumably because it was thought unlikely that there would be another HMS Whitley). I suspect that as the original cast of the destroyer's badge was in existence, someone re-used it, but inserting the new name.
p.s. the Speaker did get a new motto - the Whitley had used 'silence is golden', but the carrier was given 'faceta non verba' (deeds not words)