I like how immediately we're thrown by these opening sentences into frantic action! I have a few thoughts that may or may not he helpful. Not better ways of phrasing things, just alternatives to think about. Take them with a heaping of salt. I'm just trying to visualize this scene.
Given how critical fire and wind and home are, could you bring one or more of them further forward in the first sentence? Like "The wind blew steadily for two days from the north, then swung to the southwest and drove the fire toward our mountain home."
Are the cardinal directions even important, given that we don't know the lay of the land? You might say instead, "For two days we thought our home in the mountains would be safe from the fire. Then suddenly the wind swung round and drove the blaze right toward us." Or bring the fire forward, like "For two days the line of fire seemed distant from our home in the mountains. Then suddenly the wind swung round, driving the blaze right toward us."
You use the word "into" in two places where "in" might sound more natural: "strapped baby Kelly in her car seat" and "so the golden retriever could jump in."
"Flakes of ash swirled around us" might be more vivid if the ash is directly connected to one of the people, like "Flakes of ash were blowing onto Miranda's dark hair (or shirt) as she strapped baby Kelly in the car seat."
The line about "Take the river road to your mother's place" makes it sound like the speaker is being left behind. Is he or she not leaving with Miranda, baby, and dog? If the speaker is leaving, they'd say, "Let's take the river road to your mother's place."
I did pause to wonder about there being a river in the mountains. I don't think of a mountain as having a river—usually smaller bodies of water like creeks and streams going downhill. Are they right next to a valley with a river running down the middle of it? If so, wouldn't there a be a connecting road going down the mountain to that valley?
The final words of "small space allotted to her amidst hastily assembled possessions" has a lot of sibilance: eight S sounds, the last three of them double S's. "Amidst" and "allotted" and "possessions" all sound rather formal for so urgent a situation. What about something like "I opened the hatchback so our golden retriever could jump in between (or next to) the few things we'd quickly gathered. 'Let's take the river road to your mother's place.'"
The more I think about the sequence of events, the more it seems to me they'd have already discussed the route before leaving the house. "Oh no, the fire's coming our way" then "Let's go down to your mom's place." It doesn't seem like the decision about where they're going wouldn't be happening until stuff and baby and dog are almost ready to go.
But I don't know what's happening next (like is the speaker deciding to stay behind at the last minute, to spray water on the house?), so my image of the sequence could be off...