I can and do, write tests for my code but I want to make a decent effort at the above question.
There are some smart people in here. How would you go about this?
That said, you mentioned mainly browser. One way to estimate the cost of testing (regardless of domain) is to consider what your code coverage needs to be, figure out for your language and tooling how many lines of code that is in comparison to the code it is testing, then calculate a cost based upon how long it takes to create said tests. It won't be perfect, and I'd not highlight the line of code methodology necessarily to management but it is an effective way to understand the problem. e.g. if I wrote a 2000 line module of moderate complexity and I had to write 3000 lines of test code to get to 100% coverage then my testing will likely be nearly an equal or slightly higher cost then the development, again a lot depending on tooling where code generation might help reduce the workload.
For example, I work on medical devices and associated products (including browser based). The general rule of thumb we are finding is that the cost for testing devices is roughly 1.5x the cost of the initial code creation. Said another way, if it took us 1 month to write the initial functionality, we expect it will take us 2.5 months to write the initial functionality with proper test coverage and test it all. Move to the web, node/javascript in our case, and the tooling is pretty slick and will create a lot of the boiler plate for you, so the testing there is more like a .6-.8 to 1 ratio. Meaning for every month worth of feature creation I generally estimate it will take 2.5-3 weeks of test code, so we would book 1.75 months for development & test creation for the same 1 month of features.
Mind you, this can all vary greatly depending on the team, requirements and needs. But should at least help you figure it out. For me when asked for a quick "guess", I always ballpark testing as 100% the cost of feature development for most projects. As an example, I recently was involved in rewriting some API's for our solution, we have spent as much time writing tests and testing as we did creating the initial features.
You will almost always hear pushback from people that don't understand that testing costs real time and money, but a key to it is if the testing is setup and done during feature development the cost is incremental to schedule and provides robust regression analysis and as you add features it gets more robust and takes less time to deploy a feature because testing will help validate the quality of the code.