Good Morning!
Is your old website in WordPress? If you are considering building a new website in WordPress, I PERSONALLY would just as soon start building a responsive theme. Or even put a responsive child theme on individual pages to hold its place until you have something ready.
We don't totally know how much any of these algo updates from Google impact us until after the fact. Everything is speculation until it actually launches, Hindsight is 20/20. You will also see impact from making changes to your layout, website organization, etc.
Making your website responsive means the least amount of change overall. Its very hard to answer and give suggestions on what to do with another persons website, especially without having looked at the site at all. There are so many things to take into account, such as timeline for a complete overhaul, the impact of a incomplete mobile site on sales etc.