Can you answer Yes to these questions:
Are you in a niche role that is hard to recruit to?
Were the missed emails due to you focusing on higher priority work and people were chasing sooner than needed?
Have you been highly performing on every other day, outperforming your kpis and getting great feedback?
Are you in a role where you can't work at less than 100% (and I mean your employer can't have anyone there at less than 100 due to risks of high risk mistakes, not that you don't feel up to it) or if contagious due to mixing with very vulnerable people?
Was your lateness due to a childcare issue that was unavoidable, non repeatable and was one event that happened 4 days in a row e.g childminder ill and substitute childcare couldn't take your child any earlier?
If you can answer yes to every single one of these then you may be ok. If any of these are a no then I think you need to accept that you won't pass probation and that you have an unrealistic view of workplace expectations.
Btw the new laws don't give you a free pass to take the piss as probation periods are still allowed.
For context, my team are not expected to work when ill, they have flexible working for childcare, and I wouldn't even know about the odd missed email unless it was a really important one. What my team do though is bring in millions of pounds of business every month, work long hours in a stressful role, support each other, and have exceptional skills that are not easy to replace. The flexibility works both ways. You seem to want it all one way.
Edited for typos