When can I expect to be billed for the duties and taxes if I ship an international package using a DDP service?

This depends on the customs office of the country you are shipping the package to. Some countries are able to bill that immediately, and you should see those charges come through on the same invoice as the shipping charges, but other countries that aren't as responsive can take weeks or even months to respond. It is rare for those charges to take longer than a month or two, but it can happen.