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Here's a video (quite lengthy w/multiple parts), while not exactly your proposed scenario however is still interesting and insightful.
https://youtu.be/zfaYrTEg7hg
But the one in the game seems to be a small splinter fleet. So perhaps 10 million would be enough.
But if they would stand any chance against Tyranids depends more on the battlefield, the tactics used, the equipment available and the quality of the guard in question.
The thing is, you can't really win a war of attrition against Tyranids (at least not on a world filled with biomass like the jungles in the game), they keep consuming biomass (including their own and your dead) and use that to spawn more Tyranids.
You need to get rid of key beasts like the swarmlord and the hive ships. Basically you need to get rid of their connection to the Tyranid hivemind. Space Marines are ideal for stuff like that (decapitation strikes, actually boarding a hive ship), but its not impossible for guardsmen and the imperial navy (shooting the ship rather then boarding) to handle alone.
Even then you will likely have what can be called a Tyranid infection on the world forever unless you burn the whole biosphere down to the ground and start over.
Doable imo, we saw the hive fleet's composition and it is a small hive fleet with just a few big ships.
And whilst this is for game reason, lore wise the astonishingly low diversity of the tyranid organisms we encounter coupled with the utter absence of any really big tyranids (hieorotitans, etc) gives further credence this is a baby hive fleet that isn't able to deploy anything bigger then carnifxes.
There is only the Cadian 9th as the imperial guard, it's stated as much in the game.
What that means number wise? Your guess is as good as mine tbh since yeah 40k writers are hopeless with numbers and scale.
I'd put my chips on the cadian 9th having 100k men or so when they started the campaign, and down to 50k or so by the time of the game's events. Doesn't sound like a lot but let's assume the many millions of pdf who should be there took the lion's share of the casualties which makes it particularly stinging that titus praises the cadian 9th for surviving so long whilst casually ignoring the countless pdf who died for said cadians.
I don't think its stated whether this is a splinter fleet or a major tendril, though considering the potential threat the Imperium can't exactly just leave them to their own devices.
No character calls it a small hive fleet, but we as the audience know it is when we are shown a dataslate showing said hivefleet. I believe it's in the first meeting between titus and the cpt? Or it was shown in the intro, either way it's at the very start of the game.
This hive fleet only had a handful, like 4-6, motherships, the rest of the fleet, about 100 if even that, were all small fry.
Your average imperial battlefleet could easily deal with this, but apparently there was none available, presumably dealing with far bigger and more dangerous splinter fleets.
If you cut their force in space and avoid them making landfall, you probably don't need hundreds of million of guardsmen.
In fact, this is where actual Spess Muhreen shine, it's in the name, muhreen, they can board or deep strike (Terminators) into enemy ships to cripple them before they get near the planet.
True, but OP said "instead of Space Marines coming to help" so I assume that means it's just 10,000,000 guards vs the Nid splinter fleet, no? If that's the case, the guards lose.
If there's no intervention from the Ultramarines or Chaos, (chaos killed quite a few tyranids in the campaign as well) and it's just 10m guards against that splinter fleet.. I don't think the guards hold out very long. Laser pointers vs Carnifex tends to not end well for the imps..
The Blood Ravens needed 40 some odd ships to stop a splinter of Leviathan.
The T'au had to work with the guards to stop the splinter fleet Gorgon, and those Nids still bulldozed a few dozen worlds before being destroyed.
Although if it had been a "true" hive fleet, like Leviathan or Kraken.. as Scrapper said.. you're talking hundreds of millions of guards to the grinder.. and even then, probably lose.
Relevant series for those interested on the topic: an ongoing series chronicling a hive fleet invasion along with rare demonstrations of what the pdf do to defend their planet until the imperial guard/space marines arrive.