Assuming this is an official UK government petition, 10k signatures means HMG has to provide a response. This will be particularly anodyne, and will merely quote current policy - so expect words like "understand disappointment", "working to keep the UK safe from Covid", "protecting our borders", but absolutely no change in direction.

100k signatures secures a debate in parliament, but this will likely be scheduled so far in advance that the situation will have already resolved itself.

If it's hosted on a site such as change.org, the government will likely ignore it completely.

To put it in perspective, a petition to revoke article 50 - the mechanism for leaving the EU - received over 6 million signatures. It essentially had little to no effect. Your very best chance is that SA moves from red to amber, but as I detect considerable resistance to getting vaccinated I very much doubt this is imminent.