General aviation pilot reported a traffic conflict with another aircraft during climb on an assigned heading from ATC. The pilot did not see the other aircraft and ATC did not identify the aircraft in the traffic pattern.

Narrative

With IFR clearance I departed FDK Runway 05; runway heading and when instructed turned to heading 290 degrees. At 2 north of FDK (departure control announced this when I checked in); at my 11 o'clock and 50 to 100 feet below was a small aircraft that I interpreted to be on left downwind for Runway 05. From runway heading; Tower advised me in the same transmission to both turn left to 290 degrees and contact departure. Departure was busy talking with the other aircraft; and I had to call 2 to 3 times to connect. Just as Controller identified me and told me 2 miles north of FDK; I noted the other small aircraft. There was no mention to me by the Tower or Departure Control of conflicting traffic. Neither my TCAS nor ADSB showing conflicting traffic. While I was on Tower frequency; I do not recall anything about traffic entering or on downwind. Reported visibility was 5 miles; but looking west it was only about 3 miles (for me and presumably the other small aircraft). I do not think that FDK Tower had radar information. The FDK Tower operator seemed over tasked.