If I'm applying for a trademark in several countries, would a refusal in one country have any implications for other trademark applications?

The simple answer is no. Even if you apply via the Madrid System, all applications are assessed independently by the respective IP offices.

Imagine a scenario where you apply for a trademark in India, the US, Japan, and Canada. You receive a refusal in Canada. This will temporarily stop your Canadian application until the issue is resolved; however, it won't have any effect on the rest of your applications. Your Indian, US and Japanese applications will continue in the process.

Let's look at a different scenario. Imagine you applied via the Madrid System, used your US application as the base and designated India, Malaysia, and the UK as contracting parties. If an issue is then raised in India, it won't affect the rest of the applications. However, if your base application in the US fails, your WIPO application automatically fails as well.