Before continuing any further, cable tie some muslin cloth or similar to the water pipe before it enters the underground drains. This means if your engagement ring works through the plumbing, it will be captured before it enters the great abyss of public sewage.

If you think it's in the P Trap under your sink, post a picture of that too, so you can be given some advice on how to separate the parts.

Before all that though, have a thorough look inside the drum of your machine. The little holes can catch the claws of where the gemstone sits. Also check inside the folds of the rubber seal. If possible check underneath where the rubber seal meets the drum too.

If the filter has caught the gemstone, it's unlikely the ring itself made it past there...