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First, one cannot definitively identify a flathead from the heads alone. All the heads essentially interchange. With that out of the way, let's play along and presume that this would be a 1952 or 1953 engine. 1953 had a lot of one year only components to it and not designed for a first generation F series whereas the 1952 engine would essentially drop in to a 49 configured for a V8. Of course, you'd be wise to make sure the engine block has all the truck accessories as well.
Now, I believe you already have a flathead motor installed that you are happy with, so ignoring the motor swap completely, it seems you want to take the transmission. However, again, the 1952 vs 1953 light duty Ford column shift transmissions are different. And then you want to take that Ford column shift transmission and somehow convert it to a center floor shift [1953/55] Warner light duty with overdrive?
Just get a 1953/55 Warner light duty with overdrive and any appropriate bell housing/adapters and install it if that is what you are after.
If the OD in question is a B-W OD, as was available on '53-on trucks and also was used on flathead Ford and Merc cars, the OD solenoid falls smack on top of an F-1 trans crossmember. They can be made to fit, but crossmember work is needed. If the engine/trans you are looking at is from a car, the clutch will be different too. There are a fair amount of electrical controls needed to control the OD, if they are not included they can be hard to find.
This is it about an hour from my house 61 miles, it’s $500. If it’s not an easy bolt in, I’ll keep moving forward with T5 I have. If it was an easy bolt on I figured why not. I have two flat 6 motors to replace the 8 if I need another motor. A 226 and 254, along with an aluminum head, dual intake and split exhaust.
I'm getting earlier passenger engine vibes, and that it isn't a true "EAB" engine. Also, I'm not sure I'd buy into the idea that the "rebuilt" means anything. As @ALBUQ F-1 cautioned, I don't see all the OD components, including the switches and whatnot.
Lastly, more about stuffing an overdrive into an F-1:
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