When using the remote to negotiate the screen, make selections, or type need to wait 20+ seconds between each entry. This just started happening with our new Internet provider who suggested we contact you.
That does not sound like normal behavior. You didn't mention what model Roku you have, so we can only consider the basics. You wouldn't expect a new router to cause any issues, but depending on the WiFi channels it's broadcasting on one never knows.
First thing I think I would try is if the same sluggish behavior appears when using the Roku app on a phone or tablet. If that works fine, then the interference has something to do with the remote. I'd first replace the batteries to eliminate that potential issue. Depending on the Roku and remote you have, it might be an IR remote, which requires a clear line of sight between the remote and the Roku. Strong light can interfere with IR remotes as well.
Dan Ultra (4640), Ultra (4670), Ultra (4800), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Nvidia Shield, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.