At
Skadden, they screen applications on a rolling basis and any that meet the benchmark are given a telephone interview. This is just to manage their workload, however, and applicants who apply on the deadline day will not be put at a disadvantage and their applications will be screened in the same way. That said, the earlier you apply, the earlier you will be able to reach the next stage (i.e. if you apply near the deadline, you'll likely have to wait until January to find out about TI's, but if you apply earlier, you may be able to do a TI earlier).
I can't help you with
Reed Smith and Simmons, but hopefully this helps with
Skadden!