Many US law firms tend to use the old-fashioned standard application form / resume + cover letter route without any online tests, e.g. Shearman & Sterling, Faegre Baker Daniels, Fried Frank, King & Spalding, etc. That's of course not a universal rule, but from my experience they are the ones that do not bother with psychometric testing so I would suggesting looking here, especially if you are interested in working for US law firms.
As for the speed of response, this largely depends on whether the firm recruits on a rolling basis or not - if the latter, it is unlikely they will respond to you before the deadline. Anecdotally, I once made an application to such a firm shortly after the application window was opened in October and only heard back from them again in June - when I got the invite for an interview I already totally forgot that I made such an application
If it is a rolling basis, the average time of response varies between a few days to one month (with a caveat that if you are waiting more than 3 weeks it is most likely a rejection - but again, not a universal rule!)
Hope that helps a bit