This couldn't be avoided unfortunately. I almost cut the tip of my thumb off a couple of days ago. The plaster is the only thing holding it together. Although I haven't bought any more I used the ones we already had at home.
Then I cut my second finger on the inside of a can at work. It started bleeding immediately and I was busy so had to bandage it asap hence the second plaster. Some things really can't be avoided.
For next time I will look at using cotton which can be composted. As for tape, they often have a synthetic plastic adhesive.
Apparently the plant known as Lamb's Ears is a good replacement as it is naturally antibacterial and absorbent. Although I don't see myself carrying that around in case of emergencies.