I'm assuming that your business analyst is a subject matter expert - they understand the organization and the industry that the organization operates in extremely well, and they probably have a very good understanding of appropriate laws and regulations that apply to what the organization does.
I would expect a developer, especially one who has been with the organization for a while, to have begun to acquire some of the skills of a business analyst. Perhaps not necessarily how to use the tools they do, but at least the domain knowledge regarding the organization, the industry, and regulations. However, I wouldn't expect any developer to be a drop-in replacement for a business analyst, must less have the ability to perform work and have a higher productivity than a business analyst.
To me, the timetable sounds like a major project risk. In order for a developer to function as a business analyst, there must be time allocated to become much more familiar with the domain as well as the tools used before the developer can function at the same productivity as an equally dedicated and knowledgeable business analyst.
I would recommend sitting down with management - explain what you would need to do to become as familiar with the environment as the business analyst who left and estimate the time that it would take you to reach that level of familiarity. Only after that time can you do the work.