The longest part of the process is usually the insurance authorization. Authorization time frames vary depending on each insurance provider's medical policy. On average, an authorization can take 10-15 business days, but can take longer towards the end of the year, when more people are trying to take advantage of having met their annual deductible or use up money in their FSA account.
Once the process has started, the best thing you can do is respond to all communication from Tandem and/or your distributor as quickly as possible. They may be waiting on specific information before moving to the next step in the process.
Once authorization is complete and any applicable copay has been received, shipping usually only takes a few days.