In which country are mechanical engineers in demand and paid heavily?

Okay, let me have a go at this.

Mechanical Engineering is a wonderful profession to choose and it is easily adapted to other branches of engineering.

The question is, if I may say, perhaps phrased incorrectly. Mechanical engineering is more industry specific than country specific. For example, Mechanical Engineers are in huge demand in the Oil and Gas industry. This is a worldwide industry but the specialities are specific to centres of the oil business. That being said, almost every region where oil is drilled there are Mechanical Engineers.

Again, pay is relative to industry and to an extent country. Oil people tend to be paid more than other industries. There is a salary survey which shows how engineers are paid, for a set role in the oil industry across countries and that is revealing. The US is probably the better payer but Europe is not far behind. This, by no means, makes other areas poor payers though. This is because the cost of living in many other countries is not as high.

I hope this helps you somewhat. Others will have better answers.