Green’s Dictionary of Slang

to shake hands vigorously, as if wielding a pump-handle; thus pump-handler, a handshake of this nature.

1879		‘’Arry on Crutches’ in Punch 3 May 201/1: Yaller ulster and elbers well-crook’d on the high perlite pump-’andle plan.
1901		Bulletin (Sydney) 21 Sept. 12/3: Mayoress Gillott has pulled round and is in harness again; she feels strong enough now to pump-handle the thousands who will turn up to her ‘Come one – come all’ reception, so the ordeal is fixed for Thursday next.
1952		‘Josephine Tey’ Singing Sands 221: ‘You look like a bridegroom,’ Sergeant Williams said in great satisfaction, pump-handling Grant .