For me, it would depend on how well the template is built. In general, I have a policy against using templates at all. I work for a specific degree of quality and, more often than not, cannot provide the client with a high quality outcome if using a template.
Historically, I've used a template once and, unfortunately, it didn't turn out so great in the long run.
My suggestion to you is this: If you can find a template that is relatively easy to work with and is easy to modify to the client's needs, then go for it. Otherwise, the deadline should be moved - the client should try to understand the position of their designer/developer, and set dates accordingly.
Side bar: In a normal situation, I start my sites from scratch, using a flex-box grid framework, and other shiny toys (like lesshat) that speed up design and development. If, however, I have a project with a tight deadline (this has happened to me once), I tend to use Bootstrap and a starter theme that's easy to expand on.