I've built this kind of system before for more than one company. What you want to build is a continuous integration system (and this is what my company specializes in). This allows you to trigger tests each time a developer submits code. The test you describe would fire up several environments in parallel (i.e. one for each browser) and you would get high level status reports about test coverage and success.
Don't hesitate to contact me and I'll do my best to walk you through the process via phone, email, and how we see fit.