How Were The Lozenge Shaped Plugs "Put" Into UK Tin Farthing

Since the copper plug was primarily an anti-counterfeiting device, the answer is "quite laboriously", I would imagine. And, as it was an anti-counterfeiting measure, I don't think they made their methods public knowledge. I think it's safe to assume that they punched a hole in the planchet, manually inserted the little square piece of copper, placed the whole assembly into the coin press and squeezed. As with modern bimetallic coins, it is the pressure of striking that makes the metal flow around and into the gap between the two pieces of metal, making it harder for the core to simply fall out. Since tin is a much softer metal than the modern alloys used on today's bimetallic coins, the pressure needed would be much lower.