How deep of an impact could the DOL's proposed fiduciary rule have on advisors?

Like Charles, I too am subject to a fiduciary standard, so it should likely have no impact on me, my firm, or any other advisors that already operate in that world. And like Charles says, a lot of the impact will boil down to the final language used in the rule that is ultimately approved (if a new rule is ever, in fact, approved).
The ones that it may impact are advisors who work for broker-dealers or who are otherwise subject to the suitability standard, and it could also impact insurance agents who call themselves financial advisors even though, at the heart of it, they may predominately be insurance salespeople.
I personally feel that the biggest impact is going to be that if the rule is approved and implemented, those that were previously held to a suitability standard may find themselves in more positions where they have to be able to show that selling a person an annuity or life insurance policy as opposed to investing the money in mutual funds, ETFs, or stocks, was actually in their best interest, as opposed to merely a suitable offering for their situation. Or they will have to actually show why putting them in a mutual fund with a 5.75% front-end load and ongoing trailing commissions simply for keeping them in that fund is actually in their best interest as opposed to having put them in a very similar no-load fund that charges a 0.25% annual expense ratio.
In other words, they will potentially find themselves in more situations where they have to show that they are actually abiding by their fiduciary duty, instead of just putting someone into a product that pays them well and is "suitable" but not necessarily in the client's best interest.  Some may decide that it is worth it to them to operate under this new standard and change their practice, their fee structure, etc., and some may not.
At the end of the day, it will certainly have some impact, and some of that impact will be on their bottom line, which is why you have seen such strong opposition from certain segments of the industry. If the rule passes, they may have to change the way they operate, change the way they bill, create new ways to bill, etc. all that allow them to more easilycomply with the new rule. On the other hand, there are some of us who will likely see little, to no, impact at all, because we have already been operating under the fiduciary standard for some time.

What a great question!
As a Registered Investment Advisor who is all ready held to a fiduciary standard and a former wire house representative who wasn't so held, I have my jaded answer all prepared. But it might not be wise to post the unexpurgated version here. I'm not trying to treat your question lightly: I think the wirehouse advisors will be negatively impacted and it will lead to a contraction in their population. Those who enjoy the business may opt to accept the new rules, but choose to operate on their own rather than for someone else.
The big unknown here is the final verbiage of the rules themselves. Don't think for a moment that the major firms are not going to try and influence the final regulations to become effective in a format that will allow their business models to be altered.
May we all live in interesting times!