They're still pretty much keeping the prime name Paramount+. Showtime is already a part of the Paramount+ service as an add on, so they're probably just going to merge the two into one and up the overall price a bit going forward.
If anything this just means you may be saying some of the Paramount+ new Star Trek content on the Showtime cable and satellite channels at some point soon after the merger.
When HBO Max started there were some items that were Max exclusives, and thus weren't being shown on any of the HBO cable or satellite channels. That's pretty much gone away and most of the new stuff on HBO Max ends up on the HBO cable and satellite channels at some point as well.
Paramount's just going to leverage their Library of content across more of their media entities. Certainly certain things will appear on Paramount+ before they appear on the Showtime cable and satellite channels, and vice versa.
Remember that Paramount also runs a free streaming service called Pluto TV; and they already have two Star Trek channels in the US version. The first one alternates between TNG & TOS, and the second channel shows DS9 continually.