I am recently retired and had valve surgery 8 years ago and I was on Warfarin. I was an ATCO, but the rules are the same, I believe.
I had to take a reading a maximum of 12 hours before starting duty, and then every 3 days thereafter, until I finished my period of duties. I worked 6 days on and 4 off, with the last 2 days working generally being nights. To make it easier, I used 72 hours instead of 3 days. I had to buy a home testing kit and my own testing struips which were expensive (about £75 for 24, and they had an expiry date on them as well, so I bought from the manufacturer).
I had to see a cardiologist every year to check me out, and as part of this check, I had to take along my record of testing to show that it was in range. Also, I had to take it to my annual medical.
Of course, if I went out of range, I couldn't control, even though I was able to work normally. The hospital doesn't get excited if you go out of range a bit, only if a lot and for a long time. There are certain foods that you shouldn't eat, such as cranberries, basically anything high in vitamin K. Antibiotics also send it a bit wild and it does take time to bring it back in range, it's not like diabetes where you can change it very quickly, it's usually a couple of days to get back.
Hope this helps, and if I can help, let me know.
TS