The only thing different about using CloudWatch Logs in another account is the credentials you'll need to use.
I would do it like this; in your logs account, create 3 IAM users:
Give each of the users the necessary IAM permissions to write logs to CloudWatch logs. Also, create a set of AWS credentials for the 3 users which you will use to authenticate with.
Then in your application, depending on what stack your in, configure your code to use the relevant credentials (i.e. Dev, Prod) to write to CloudWatch logs. The only difference should be wherever in your code you create the CloudWatch logs client. You should pass the client the credentials for the relevant IAM user from your "Logs" account.
EDIT:
If you're using the CloudWatch Logs client, the idea is the same. You create the users and instead just give the client the relevant AWS credentials. You specify the creds for the user from your logs account in the /etc/awslogs/awslogs.conf. The credentials you use is what will send your logs to CloudWatch in the other account. The process you need to follow is basically documented in the AWS docs