Deu certo. Eu inseri o seguinte no bloco na minha pergunta acima:
hostname: mongodb
volumes:
- ./data/mongodb/data/log/:/var/log/mongodb/
# the healthcheck avoids the need to initiate the replica set
healthcheck:
test: test $$(echo "rs.initiate().ok || rs.status().ok" | mongo -u root -p imagiaRoot --quiet) -eq 1
interval: 10s
start_period: 30s
