This might work, but some things are up to the GM
You might be able to mount the spirit
The summon bestial spirit spell states:
[...] The spirit manifests physically in an unoccupied space that you can see within range. This corporeal form uses the Bestial Spirit stat block below [...]
From this we can conclude that the spirit is physically there and interactable (it is corporeal; it can be attacked as well). So now we just need to check whether it works as a mount. The rules on "Mounted Combat" state:
[...] A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount [...]
If you become Tiny (through Wild Shape most likely), then the spirit is one size larger so it qualifies there; however, it will be up to the GM whether the spirit has the appropriate anatomy to serve as a mount for whatever creature you turn into.
You can command the spirit while Wild Shaped if you can speak a language
The rules on controlling a mount state:
[...] You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.
You can control a mount only if it has been trained to accept a rider. [...]
The spirit has an Intelligence of 4 but whether or not that counts as intelligent and whether or not the spirit has been trained to accept a rider will be up to the GM. Regardless, the summon bestial spirit spell states:
[...] The creature is friendly to you and your companions for the spell’s duration. In combat, the creature shares your initiative count, but it takes its turn immediately after yours. It obeys verbal commands that you issue to it (no action required by you). If you don’t issue any, it defends itself but otherwise takes no action. [...]
Thus it will at least obey the commands you issue. Notably, if you don't issue any commands it simply won't do anything, though it's a bit unclear how (if at all) that clause applies outside of combat or if the spirit is considered an independent mount.
Important to remember is that the spirit understands any languages you speak, so if you somehow turned into a Tiny beast that knows a language, you could communicate verbally while in beast form (if you somehow turned into a telepathic Tiny beast, the GM will have to decide whether you can command the spirit). In other words, if the beast you turn into does not know a language, then you cannot command the spirit.
What sort of commands you can give is up to the GM
Issuing commands is something the GM will have to decide. Can you issue a non-immediate command: "fly up in five minutes"? What about tasks that take more than six seconds: "bring me home"? Do you have to issue a command every six seconds? How does the spirit having only 4 Intelligence impact this? It's unclear how complex commands can be and what their requirements are, so it'll be up to the GM.
Alternatively, just get grappled
The Small spirit can grapple (as far as I'm aware) every official PC race. This would mean it moves at half-speed but it still allows for a rather sizeable amount of aerial movement. (60 feet per turn using the Dash action, and you have 1 hour so that's roughly thirty-six thousand feet covered). This avoids practically all questions about appropriate anatomy, controlled vs independent mount, languages spoken, and command complexity while also letting you keep a use of Wild Shape.