the answer depends on the country you are applying to. While in US and UK consulting companies are more open to various types of backgrounds (one of the people I helped to land an offer at McKinsey was a former pianist), in other countries such as Italy consulting companies focus almost exclusively on candidates with a background in Business or Engineering.
Generally speaking, as Riccardo mentioned, MBB companies are more open to a variety of backgrounds, also due to the higher availability of training programs. In general, I would keep options open and apply to all first and second tier firms in your region – such as MBB, Roland Berger, Oliver Wyman, Strategy&, ATK – unless they focus on a niche which is not interesting for you (Oliver Wyman for example focuses only on financial services in some countries). Given your background, you may also consider companies focusing more on healthcare such as ZS Associates and IQVIA. If you don’t manage to receive invitations now, as a backup plan, you may pursue an MBA and reapply after that.
As a strategy to approach these firms, I would recommend to have a current or former consultant to help you to prepare a proper CV and Cover. Ideally you also want to find referrals for your application – at the link below you can find some suggestions on that:
Hope this helps,