Gospel for this post is "Hitler's Luftwaffe" (Tony Wood, Bill Gunston - Salamander Books, 1977). Given the publication date, I'm happy to accept correction from more recent sources.
Grey Havoc asked what would have been ready for Spain, if not the Bf 109?
According to the first part of the book, which takes the form of a multi-chapter written history, the initial dispatch of the Legion Condor was He51B fighters and Ju52/3m transports (some of which I understand were modified as bombers). Bf109B fighters, Do17 and He111 bombers joined in early summer 1937, with the Ju 87A and Hs 123 dive bombers coming in later in '37.
The second half is the airplane-by-airplane technical section, arranged by manufacturer. Flipping to the Heinkel section, we find the He 112 on page 185 of my edition, 17 (out of 30) of which are described being flown in Spain "by volunteer civilians" and being preferred to the Bf 109C, sufficiently so that the Spanish kept them until after WW2. Although it doesn't say when the -112s arrived in Spain, the B-series production prototype is described as flying in May 1937, so perhaps figure a couple of months to build and test-fly the relevant numbers?
The He112 gave 317mph with a 680hp engine.
The Ar80's specs are given as max speed 217mph, range 497mi, ceiling 32,800ft. This puts it in roughly the same performance bracket as Supermarine's Type 224 fighter. However, it's listed with engines between 525hp and 695hp, and the text doesn't make clear which one was fitted for the performance figures given.
The Fw159 gave between 239 and 252mph depending on the engine power (610hp, 730hp respectively), but it was killed as a service aircraft by maintenance issues.
The Bf109B gave 292 mph (SLOWER than a Hurricane I, please note!), from a 635hp engine.
In terms of straight line speed, the -112 is the tear-away winner with the -109 out of the picture, and the others don't even come close. I suspect the work required to turn the Ar80 into something competitive is closely akin to the work required in turning the Supermarine 224 (which it reminds me of) into the Spitfire - the engineering and design lineage can be traced, but in the end they are completely different airplanes.
That covers the fighters.
Turning to the Zerstorers, the Fw57 is described as being grossly overweight with poor handling, 251mph on 2 x 910hp.
The Hs124 offers 273mph on 2 x 880hp radials, and seems to have died a death as a result of the evaporation of its basic concept, but with baseline performance like that (and the promise of much more powerful engines along the way), it might have made a good tactical bomber.
The Bf110 we know all too well. Remove it, and the Hs124 seems to be the better contender - if only on the basis of significantly superior straight-line speed on lower overall power (implying better aerodynamic refinement and a less overloaded airframe, I guess).
The real dark horse is the Fw187 Falke; A-0 variant first flies in February 1939, too late for Spain but early enough for Poland - the V6 touched 390mph on 2 x 1000hp, but the A-0 is credited with closer to 326mph with military equipment on board; 2x20mm, 4 x MG. The RLM seems to have had a hate-on about this aircraft, because they were sent to Norway and preferred to the Bf110, but when this was found out, promptly recalled. Messerschmitt and politics, I guess.
So there's some technical food for thought. Long story short, it seems the real-world Legion Condor doesn't start to be competitive until the Bf109 arrives, and then it becomes VERY competitive. Eliminating the Bf109 means a delay of, shall we say, six months before the He112 arrives in numbers, but even so, that still leaves two years for the Luftwaffe fighter arm to cut its teeth before the invasion of Poland and so it doesn't change history much from that perspective in isolation.
Whether it's unbuilt projects, promising prototypes that saw successful combat trial and then mysteriously went no further, or what-have-you, personal politics enters so much into Nazi German combat aircraft development, it isn't funny. It's rich ground for a novel, simply because it offers a potential window (if you so decide to write it that way) into a bunch of morally corrupt backstabbers making the worst decisions for the "wrongest" reasons at every turn.
Taking one prominent person out of the equation at any point shifts the dynamics so radically it's impossible IMO to determine exactly what would have happened. Real Life took out Walther Wever, who might have backed the Do19 or Ju89 and/or proper development of the He177. Put him back, and any one of those three bombers suddenly takes a much more prominent place. Arguably his successor Kesselring was right about the air force that was needed to conquer Continental Western Europe in the first part of the war, but after that (it could be argued), the broader vision was lost and Germany paid the price accordingly.
Looking at these contenders, though, the Do19 seems no better than a four-engined development of an A-W Whitley-class bomber (max speed less than 200mph) while the Ju89 seems by far the better bet, but still only in the 250mph class in prototype form, so who knows what the development potential really was? Conversely the developed 4-engined 177 variants both touched around 350mph, and although they either didn't fly during the war or flew too-little too-late, offer a window into what might have been achieved if development had gone that way from the start.
Ultimately if you take Messerschmitt out, I think Heinkel and Focke-Wulf fill the gap - Heinkel in fighters until the Fw 190 is ready and then switches over to the He280 and quite possibly the four-engined 177 (Messerschmitt was a good Nazi, who might have been undermining Heinkel in more than just the fighter field; with him out of the way, the "He-177/4m" might have stood a better chance), while Dornier and Junkers stay pre-eminent in medium bombers and Henschel specialises in low level attack and army support. That's a broad brush, but it should at least give you a general landscape to work with.
Of course if you REALLY want to alter the course of Luftwaffe technical developmental history for what might be considered the better (or for the rest of us, the worse!!!), the people you need to kill early are Udet and Goering.