Is "real-time" a term known to every English speaker?

Even in software engineering it's not widely known, or even uniformaly defined. To embedded systems people it means that the result must be computed in a certain time or it's useless - to other people it simply means that the answer is produced 'live' or essentially immediately rather than offline

Maybe I'm a little geeky, but I understand what real time means. It is not a synonym for fast. First let's agree on what it means:
OED noun the actual time during which a process or event occurs
[as modifier] (real-time) Computing relating to a system in which input data is processed within milliseconds so that it is available virtually immediately as feedback to the process from which it is coming.

In your case, you could gloss immediately, at once, right away for the adverbial phrase in real time: the algorithm needs to be fast to produce the results "in real-time" (right away, at once, immediately).

As I understand it, "Real time" means that the system is always guaranteed to produce the results before some arbitrary deadline (see Wikipedia). What that deadline is however, depends on the system. It can be microseconds, and it can be months. So "fast" is not really appropriate here.

I suggest you use "real-time" in your text, but put a short explanation in parentheses right after the first time you use it. Something like:

The analysis package will produce results in real-time (before the enzyme has broken down), so it will be possible to administer the antidote before the patient has died.

As the other answers have shown, this is not commonly agreed upon, even among those who know what it means. Within IT, it means that the data is delivered to the users screen as soon as possible after it has been acquired. Their display should show data as it is now, and is a very challenging requirement, even though the specifics ( exactly HOW real-time they want it ) may end up being discussed at length.

In your case, I would be tempted to say "time-critical" rather than real-time. So "the delivery of the results is time critical, so this algorithm which is part of that is also time-critical" - which is less specific, but indicates that it needs to take place as fast as possible without losing accuracy. It could still be valid, if it were to take 1 hour to run, if that is the fastest is can manage. That is not real-time, but is still time-critical, potentially.