Take a deep breath and try to approach this with a clear head.
When you say they are not happy with your processing speed - do you mean your Internet speed, or how fast you are processing the work?
Go and work out NOW a couple of options for tomorrow:
- Go to the office
- Use your phone as a mobile hot-spot for data (work out how)
- Go to a local cafe, library, similar
- Ask a friend/neighbour/mum to work at theirs
Then have a cup of tea.
Next time, be honest with your manager and do it early. ("I'm having a problem today, the Internet service is completely down. I have done a, b about this so far and am about to try c. If that doesn't work I'm not sure what to try, can you help with any ideas?")
Pro-actively check in with your manager a couple of times a day. Try to find opportunities to be constructive in your interactions.
Ignore your ex husband. What he thinks is neither here nor there.
In general, businesses are struggling to find staff at the moment and it's probably in their interest to bring you on rather than fire you.
If it does all go wrong, you've got a few weeks of experience. Learn from it, put it on your cv & move on.