An Air Carrier Captain reported they descended below glidepath on an approach in IMC and received a GPWS alert and Low Altitude Alert from ATC.

Narrative

Aircraft X ZZZ-DCA on departed late due to a Maintenance delay with tail swap in ZZZ2. Right at pushback; ZZZ Ground said DCA had a ground stop. We cancelled the push. The ground stop was for 'traffic volume' and the update was an hour later at XApm EDT with a low chance of extension (30%). We opted to keep all the pax onboard in case the ground stop was lifted early. We double-checked the weather; which looked rainy but otherwise benign (no alternate required). However; we were in a Aircraft X so we hand-built the LDA Y 19 approach for DCA and realized we could only do that or the RNAV 15; but only if Takeoff and Landing Data was good for Runway 15 (wet). When the ground stop was lifted; we got clearance to push and were given a wheels up time of XA:13 pm. Around our TOD; we heard Center tell another aircraft there were limited approach options available in DCA. Even though the DCA ATIS was advertising the LDA Y 19 approach (as they were when we departed ZZZ); the only available approaches were RNAV 19 or 15. We had good numbers for 15; but with a wet runway we required thrust reverser credit and only had about 400 ft. to spare. We opted for me to fly given the slim margin for error. The weather at the time was winds around 10 knots from 100 degrees (left quartering headwind not accounted for in our TOLD); 8 sm visibility; and ceilings at 1;000 feet. Being in an Aircraft X; we could only fly to LNAV minimums which required 2 & 1/2 sm visibility and would get us down to 836 ft. HAT. While on downwind a PIREP stated the ceilings were at 500-600 feet. We told Approach we would continue (with the required vis) but may end up having to go missed since we were only LNAV capable. We sent a message to dispatch to ask if IAD was a viable alternate. We were told no due to congestion; but RIC would work if fuel permitted. We quickly checked and we did have the fuel to go to RIC if we went missed. We were cleared direct NIPEE and cleared for the approach. Once inside the FAF; Tower asked us if we could increase speed momentarily by 10 knots (125 to 135). We accepted the speed increase (presumably for Runway 19 traffic sequencing). At the FAF I had set a VS on the FCP of 800 fpm; intending to reduce it to 700 fpm. However; I became fixated on the speed increase and dropped the snow flake out of my cross-check. Around 100 ft. above minimums (950 ft. MSL); we received an 'obstacle' caution from GPWS. It did not occur to me that I may have descended too quickly. I just thought: it's only the caution; we can continue. I then told the DEC that we could continue to 800 ft. MSL (published MDA) to try to get the runway in sight as we were still IMC (yeah; bad idea). Right after that we received the 'obstacle; pull up' warning and I executed a standard missed approach (as we had pre-briefed) instead of the proper full escape maneuver. We diverted to RIC and the DEC landed without incident from an ILS approach. We were parked on the de-ice pad (allegedly due to no available gates) and were told by RIC ops that we would be refueled quickly so we could depart to DCA. We were fueled quickly; but the weather in DCA had not improved. Also; dispatch forgot to include passengers in the new flight plan/TOLD calculations. Due to DCA Runway 19 closing at night for construction and our need to land on Runway 15 wet (plus fuel for an alternate); we would only be able to take 6-11 pax (depending on contingency fuel and the alternate location). The DEC proposed we fly directly to IAD instead; so we could take everyone. Dispatch liked the idea and elevated it to Flight Ops management for approval. We received the approval; but IAD said they still could not receive another aircraft. Dispatch directed us to RTG in RIC to deplane. It would take over 30 minutes before we were parked. The passengers were on the plane for five hours since they boarded in ZZZ. Apparently; RIC ops did not have any comm with the rampers (who said they could have parked us much sooner hadthey known about us). Dispatch wanted us to reboard just 6-11 pax to fly to DCA; but we were about to exceed 15 hours of duty. Needless to say; our passengers and our crew were completely devastated and exhausted. After deplaning; it took over 90 minutes to get a hold of scheduling and get hotel rooms. We arrived at the hotel around XG30 for a XO30 report the same day. The low altitude alert from Tower was too late (concurrent with GPWS warning). Even a few seconds earlier of Tower saying we were below profile could have helped.