I believe Mailgun send attachments as a multipart file.
For each attachment there should be a url similar to https://api.mailgun.net/v2/domains/<<<YOUR_DOMAIN>>>>/messages/<<LONG_STRING>>>/attachments/0 that conatins the file. Note you'll need to use your Mailgun API credentials to access this url as it is password protected.
Mailgun allows you to attach custom keys and values to each message. See https://documentation.mailgun.com/en/latest/user_manual.html#attaching-data-to-messages
Why not use a key titled something like attachment-NUMBER with the value being the full path http://site.co.uk/file.png (or say the AWS s3 path).
Have you tried bin.mailgun.net? It's very useful for web hooks which are annoying to debug. See http://blog.mailgun.com/product-update-new-postbin-for-debugging-webhooks/