Regarding COVID-19 response, we have the stringency index. But I am wondering is there any source that documented the first law implementation date or month regarding society restriction (social distance, school closures, workplace closures, and travel bans) in every country.
I think the data you're interested in can be found in the source Our World in Data uses for their index calculations (this is the website of the researchers dedicated to the dataset, and this is the github page with all the data downloads.) Cross referencing that dataset against major news outlets' Covid timelines (for example this one from the NY Times or this one from the Washington Post) should be enough to get a pretty good idea of the dates of lockdowns and other major milestones. (That dataset has been used in other peer-reviewed articles, like this one in Nature and this one in the BMC, so at least it's been used successfully in a peer-reviewed context.) The only other comprehensive datasets I've seen are this open-source one and this one generated using AI-assisted tracking. I know much less about those two, so I'm not sure if they're useful and/or accurate.
Alternatively, there seem to be a few other published papers that look at the effectiveness of different government measures that don't seem to reference the BSG dataset. This one in the Lancet says their data was collected from public sources and is available upon request, so it might be useful to shoot them an email. Others (like this one in Nature) aren't that explicit about where they gather lockdown data from, but don't directly cite the BSG working paper, so they might have additional data sources to potentially ask about.
If you're looking for more datasets, they're usually labeled as measuring "non-pharmaceutical interventions."
The last thing I'd mention is to offer a word of caution regarding modeling the effects of lockdowns (and other NPIs). The specifics surrounding the context of lockdown implementation is far from uniform across countries, which can have substantial impacts on estimating the impacts of NPIs. (For example, some countries imposed lockdowns after an outbreak occurred, others implemented them ahead of time. Similarly, some countries imposed lockdowns with no real warning, while in others the government debated and passed legislation well ahead of implementation.) Other research has highlighted the difficulties in using these measures for causal analysis without adjustment. Finally, certain NPIs are particularly sensitive to adoption by a population, which can lead to behavioral responses not captured by government level policies. For example, some research into mask mandates (as seen about halfway down in this non-peer reviewed article on Nature) suggests that, while the mandates might be effective in boosting mask wearing among hesitant populations, widespread mask adoption occurred before mandates were implemented, and continued after mandates were lifted, in other areas.
Let me know if any of that wasn't clear, and good luck with the project!