The answer is, it could be non-zero (some would argue it must be non-zero), but since we don't know what the probability of life emerging on Earth was, it is impossible to quantify.
This is why this question is normally turned around - if we find life elsewhere in the Solar System (and it is independently developed), then what is the probability that life can emerge given reasonable circumstances?
The occurrence of a second instance would allow us to put a number on the probability. That number would be high enough that, since the Sun is a fairly ordinary star and that Solar System architectures may also be reasonably frequent, life ought to be common throughout the Galaxy.
There won't be any drastic changes in the habitable zone location for the next few billion years - it slowly moves outwards as the Sun's luminosity increases. Perhaps Mars will warm up a bit and become habitable again? If so, there is certainly the time available (still a couple of billion years before the Sun becomes a red giant and Mars gets much hotter) for primitive lifeforms to develop, given that life on Earth seems to have come into existence very soon (less than a few hundred million years) after the conditions became suitable.
The asteroid belt and outer planets? Much less likely. The habitability or not of the Jovian and Saturnian moons is unlikely to be affected by gradual changes in the solar luminosity during the main sequence and the habitable zone remains well inside the orbit of Jupiter.
Once the Sun does become a red giant, then the habitable zone does move outwards maybe as far the Neptunian moons and Kuiper belt (Stern 2003). There are plenty of icy bodies with organic materials that might have the raw materials to form life. The difficulty here might be the rapidly changing luminosity of the Sun as it approaches the end of it's life. It will not be stable on timescales of billions of years like it is as a main sequence star. Other authors disagree on the exact location of the habitable zone during this phase, probably because of its exact definition and because of difficulties in modelling the final stages of the Sun (see below). Below is a plot from Ramirez & Kaltenegger (2016) showing how the habitable zone (between the solid lines) changes rapidly during the final billion years of the Sun's life. There is a period of about 100 million years when the habitable zone is between 5 and 15 au (i.e. the Jovian moons), and an even briefer period before that when where there could be habitability out to 30 au.
But again, given we don't know exactly how quickly life can develop, although there is an opportunity for life, we can't quantify what the probability is.
Note that all of of the above is uncertain in three ways. First, not everyone agrees on what defines the habitable zone and what temperature range it should encompass. Second, we don't know exactly how the Sun will behave when it becomes a red giant because much depends on the mass-loss rate, for which there is no fundamental theory at present. Third, the behaviour of the atmospheres and albedos of the planetary bodies or moons, which can change, will play an important role. For example, Mars is actually in the habitable zone now, according to simplistic calculations that just use the solar luminosity, and will be for many billions of years. It could just be suitable for life as we find it on Earth (antarctic microbes). On the other hand, if we demand a temperature range of 270K-300K, then Mars may only be habitable for a very brief period in the future (11.6-11.7 billion years from the birth of the Sun, according to Schroder et al. 2001) and this may be insufficient time for life to develop.
