All of the answers (jumped in and so on) are excellent,
However, I'd use the expression
I think that would be quite clever and nice. The meaning is absolutely obvious. You are instantly implying the same set of emotions and considerations as when you use the phrase "cold turkey". So, it's just like the guy going cold turkey in French Connection ... we're jumping in that aggressively, harmfully, with no thought of gently-easing ourselves through a new process..."
"We jumped in to the project hot turkey."
note CJDennis (perhaps jokingly) suggests Hot Turkey in a deep field comment above; CJ gets the eternal credit for this.
But wait ...
Here's another take. I think it would be very sensible to use the phrase
Here's the thing: the original use of "cold turkey" was by no means in relation to ending or quitting something. It appears to be used to mean aggressively-direct in any usage.
"Now tell me on the square – can I get by with this for the wedding – don't string me – tell me cold turkey."
Once you realise that "quitting cold turkey" is merely one popular use of "cold turkey," we can see that "starting a business cold turkey" or "hitting the school term cold turkey" or indeed "giving the facts cold turkey" are all very sensible and obvious.
Here's an analogy: consider the slang term "cool". One (of many) popular combos of "cool" is, say, "cool dude". Say perhaps in 100 years on a list, someone was asking "With cool dude, what would you say for _ _ _ girl?" Of course, the answer is "cool girl" - indeed cool had much broader use than that particular popular use.
Once again, particularly looking at the excellent MW early example, "cold turkey" is just another "cold" emphasis phrase, much like say "stone cold". (Note that, indeed, you could certainly say "he quit heroin stone cold".) With that viewpoint, it's completely natural to say "we undertook the enterprise from a cold turkey start", exactly meaning we jumped in with no preparation, helpful pharmaceuticals, etc.