Yes, generally speaking, you can start using your brand in commerce while your trademark application's approval is pending. A trademark is not a business license and waiting doesn't prevent you from launching your brand. In some cases, you might even want to do so. Let's look at all the considerations:
In the US, trademark rights are primarily acquired through actual use in commerce. If you filed your US application on an intent-to-use basis, you'll have to prove such use by submitting examples of how your trademark labels your goods and services, which you can only do if you start using your brand in commerce.
In many other jurisdictions, including the EU or the UK, filing an application lets you gain a priority right over anyone filing a similar trademark application after you. If your application is successful, the validity of your trademark will retroactively start counting from the filing date. This means you can start using your brand after filing because if someone infringes on your rights, you will have the means to fight back later.
However, this assumes your application will register. If you are not sure of that, you may want to hold off on using your brand. You can estimate your chances by running a trademark search or consulting a trademark attorney (we offer a free lawyer's check for this purpose). For example, you might want to ensure your trademark doesn't infringe on other existing trademarks because that would prevent it from being registered.
One more thing you can do while your trademark application is pending is use the TM symbol (™) to indicate that you are claiming trademark rights. Once your trademark is registered, you can use the registered trademark symbol (®).
