My biggest issue is, unfortunately, time. Do you have any advice on finding the time to learn when all the work still needs doing and there isn't really room in your working day to focus on automation?

This is one of the questions asked by Heleen Van Grootven for the :

AMA about transition from Manual to Automation Tester

This is the issue many of us face in their day to day life. I met couple of people asking this to me. 

Yes, that's true if we are piled up with lot of work then it becomes challenging to find out time for automation. My suggestion is to forcefully take out time. I used to follow these when I started my journey and I do follow now also:
1. Block your calendar for 1 hr daily. My preferred time was during lunch time when hardly anyone would need me for any kind of update or input. But make sure that you are dedicated to automation only. No other tasks during this hour. 
Depending upon from where you want to start - execute the existing automation, pick small failed test case and start analysing the issue etc

2. First day(sprint planning) and Last day(retrospective) of sprint is too busy so avoid those days to start sometime new. It's hard to do anything extra those days. If you set time for automation these days and fail to get time then it may lose your interest so avoiding these days would be better.

3. To learn automation, dedicate time during the start or middle of the day when you feel fresh. End of the day do not work for something new to start. By the end of the day we are either exhausted, or too much into the work that sometimes we stretch a little to finish the work. That's again losing the interest to start with automation. 

All we need is to forcefully block our time/calendar for 30 minutes or 1 hr whatever works. Then stick to it and start engaging yourselves with automation world. It's equivalent to the fact that I am super busy with my work and I am in flow to finish my work but I have a medical appointment which I can't postpone. Forcefully, I have to stop my work, come out of my flow, take a pause and go for the appointment. Once I am back, I resume my work. :) 

I hope this works for you.