Relatively recent changes in the way that Department of State handles approved petitions has all but made the K3 obsolete. A K3 requires an I-130 petition AND an I-129F. If both approved petitions arrive at the NVC at the same time (and they usually do) then the I-129F is closed, and they proceed with the I-130, which would result in either a CR1 or an IR1 visa (depending on how long you've been married at the time). The K3 still exists for those relatively rare circumstances when the I-129F beats the I-130 to the NVC.
The CR1/IR1 has a distinct advantage over a K3 - the green card approval process is included in the visa processing. No adjustment of status is required after coming to the US. You'll get a green card within weeks of arriving. A CR1/IR1 also costs less than a K1. A side affect of this is that you'll need a qualified sponsor who has signed an I-864 BEFORE you'll get the visa. If your wife can't qualify on her own, then you'll still need a joint sponsor who does. Basically, you'll be in the same boat you're in now. But wait, there's more...
You and your wife will be living in the UK. If she's earning enough at that point to qualify as a sponsor on her own then you won't necessarily need a joint sponsor. However, if her income isn't likely to continue at the same level when she returns to the US then the consulate will probably not consider it. It's reasonable to assume she'd have to quit any job in the UK to return to the US, right? On top of all this, a qualified sponsor must not only meet the income requirements, they must also be a US citizen or permanent resident who is domiciled in the US. The US government can't easily enforce an affidavit of support against someone who isn't living in the US. This means your wife will have to maintain a domicile in the US while she's in the UK, or establish domicile in the US before the affidavit is submitted, or prove to the satisfaction of the consular officer that she has taken steps to establish domicile as soon as she returns to the US (this last approach requires she arrive in the US at or before the time you arrive).
So, even though it's the same boat, it's now sailing in choppy waters.
You will save yourself considerable headache if you can find a way to resolve this while you're still here in the US. You've spent a lot of money and time already. It would be a shame to see that wasted.