In theory, the browser javascript performance object lets you do this:

window.performance.timing.responseStart - window.performance.timing.requestStart

which will return the time in milliseconds it took the browser to fetch the page from the server. This sounds exciting from the point of view of guesstimating bandwidth.

Unfortunately the reality is that at least for small pages it is nearly useless. Plotting actual measured download speed in megabits vs this browser timing metric (milliseconds) shows how useless it is:

That row of dots basically along the X axis? those have values of 1 or 2, and they are from our good friend IE 10 and IE11.

There is a small correlation, the worst bwtimer results do tend to be weakly associated with the slower end of the speed spectrum. That's about all you can really say. Don't use this performance timing object to make big decisions!