I have some Nest Protects still using the old pre-Google account and I set them up with Smartthings (classic) and the old v5 Nest Manager. Even after migrating to the new Smartthings app they still work. This option unfortunately is not available to newer users.
As others have mentioned as of yet Google have not announced official support in Smartthings for the Nest Protect, I certainly hope they do so although it likely may require me to convert finally my units to the new Google based account access.
In the meantime a complex but theoretically possible solution would be to link your Nest Protect unit(s) to HomeAssistant and then share them from HomeAssistant via MQTT to Smartthings.
I do regard Nest Protect as best in class from a hardware point of view and they look better than most others as well. However Google’s shenanigans had led me to consider alternatives - remember smoke sensors have a life span of only 10 years anyway.
I don’t know what country you are in - I am in the UK which will have different/fewer options than the US but two alternatives worth looking at are -
Both of these unlike many alternatives link multiple units together so if one detects a fire all sound off. This is a feature Nest Protect has but many other brands do not.
I suspect neither brand has official Smartthings support but again either HomeAssistant might be a means round this or you could discuss this in these forums with other users.