Let's take a look at the two abilities, and see what they say about timing.
Death Saving Throws: Whenever you start your turn with 0 hit points, you must make a special saving throw, called a death saving throw,[...]
Healing Spirit: [...] Until the spell ends, whenever you or a creature you can see moves into the spirit’s space for the first time on a turn or starts its turn there, [...]
Okay, so they both trigger at the start of the turn. We can look to Xanathar's Guide to Everything for guidance (in the form of an optional rule) for exactly this situation:
Simultaneous Effects: In rare cases, effects can happen at the same time, especially at the start or end of a creature's turn. If two or more things happen at the same time on a character or monster's turn, the person at the game table - whether player or DM - who controls that creature decides the order in which those things happen.
So, in this case the player should probably choose to have Healing Spirit trigger first, because recovering HP makes the Death Saving Throw moot. The character is still likely to be prone, so they will have to spend movement to get up, but can otherwise act normally.
If the character has no failed Death Saving Throws so far, the player could cross their fingers and hope for the 5% chance to get one extra HP out of a natural 20 Death Saving Throw. If the character does have a failed Death Saving Throw, however, there's also a 5% chance of dying instantly, so it isn't worth the risk. Besides, one HP isn't likely to make a difference against the next attack either way. Might as well save time at the table and just take the Healing Spirit's benefit and then move on with the rest of the turn.