There are people who believe that money is evil and can transform people, but I believe it is the opposite, money is a simple means that amplifies what people already are. If someone is selfish and greedy, give them money and they will become even more so.
Even if Michael Cartlidge sent the money, it was Charlotte Cox who purchased the lottery ticket. Unless Mr.Cartlidge can show proof that he and Cox had an agreement to split the prize money prior to the draw, there is no chance that he will get any amount out of it. At the most, he can file a civil suit against Cox, but even then I am not sure how it is going to help him. His only chance is in case of an out of court settlement, in which Cox maybe ready to part with 10% or 20% of the winnings, just to avoid the nuisance of legal battle.
To begin with, I believe that there was no real love in that couple. Just look at the fact that they are both overweight and don't look particularly handsome either. Surely it happened as it often happens that people get together with the one they have at hand to avoid being alone, but with people in love who plan to spend the rest of their lives together, this would not have happened.
Things as they are, I think it's most likely that what you say will happen. Lawsuits are expensive, and a settlement for a small percentage could avoid the trial and settle the matter.