I'm not OP, but I can easily see my team getting into the same situation.

Our use case involves dynamically displaying some analysis results on a status page derived from data from the field. The analysis was originally written in Python + scipy + matplotlib (Julia in OP's case). Why write the analysis code a second time just to display it on a demand-generated webpage?

You don't have to run the entire website on Julia, but maybe you do want to stand up a couple of RESTful endpoints that are integrated with the rest of the web application. If Julia can't do that, then your options will be limited.