In general the SPR is dam controlled. NFSP (bailey/foxton) runs primarily off of Roberts Tunnel diversions from Lake Dillon. It takes a big snow year for Bailey to run big from runoff. This year, NFSP will probably not run until later in the summer (July/Aug).
The SP proper above Chatfield depends on Dam releases and storage. Cheesman reservoir has a ways to go for storage, so I don't expect anything big there for a while. There's a chance that some of the high-mountain (Antero/11-mile) spill or are moved down to Cheesman which would give a few days of 11-mile and Cheeseman Canyon water. Maybe mid-late June, but unlikely.
Below Chatfield, generally nothing will run unless: Major storms rain on Plum creek and/or all Denver Water reservoirs full and start spilling.
So in a nutshell: Q: "When will we see increased flows?" A: Later, not likely to be big, don't hold your breath.