Therefore, on our second attempt (May 2008), we "bit the bullet" and applied directly through the Highly Skilled Migrant Programme (HSMP). This has some annoyingly rigorous documentation requirements, and some hefty up-front fees, but (if approved) provides "free-agent" status for the main wage earner to move between jobs freely during the initial 2yr period. More importantly, by obtaining your "own" work permit status, this frees employers from considerable cost and risk, and opens up a much broader selection of possible employment.
Note that the original HSMP program has now been closed (as of June 30, 2008), but you can still apply through the very-similar Tier-1 (General) program. I haven't researched the new Tier-1 requirements, but I'm told they're comparable to the now-defunct HSMP, so I'll speak to those in hopes they will still apply.
First of all, the most important goal of HSMP is to verify your wage-earning power. They like at least two, or preferably three or four, corroborating sources of official evidence to support your salary claims, so plan to provide ORIGINAL pay stubs, and bank statements, and W-2 tax forms, and letters from HR verifying your salary and pay history. All of these should be on original fraud-prevention security paper, using watermarks, microprinting, holograms, etc.
They'll want at least a year's worth of continuous evidence, so start now, saving every financial document that comes your way, for the next 14+ months. (Obviously, you should have been doing that anyway, "in case" of an unlikely IRS audit, but if you plan relocation to Europe, you'll have to produce the entirety of such documents, so get out a shoebox and start the collection now!)
Now, what happens if you lose a pay stub or bank statement or two? Do you have to start everything over? No, not necessarily, but it will require some extra work on your part to make up for the loss. You'll need "official" reprints from your company or your bank, and those reprints will need to be marked in a variety of ways to convince the Home Office inspectors. Generally, each should be reprinted on stock letterhead, then signed and dated by a company officer, then impressed with any seals or stamps that might be handy (ink or embossed). It wouldn't hurt to staple a business card from the signing official. Note that these signatures, stamps and seals need to be applied to every single page -- not just the first "page 1 of 3", etc. Even after all this work, the Home Office may take 12 weeks to review your application, and then reject it as insufficiently documented -- costing you a hefty non-refundable fee, and three months in your relocation schedule.
How to mitigate that risk? If you can afford it (and you really shouldn't be looking at relocation without a reasonable set-aside of liquid cash), I recommend retaining one of the many UK Immigration Specialists who will rigorously pre-review your application materials with a critical eye, making you re-do the whole thing repeatedly until you meet their high standards. This is good, because by the time you satisfy them, you'll be virtually guaranteed to pass the Home Office requirements. Being an American with a typically ingrained capitalistic faith in the motivating power of greed, I recommend immigration specialists offering "no-win no-fee" contracts, such that they only get paid if you get approved.
I ended up using FastUKPermit.com, and I was extremely pleased with their high quality of service, reliable communication, and overall diligence. They were on the low end of the price quotes I requested (still not cheap, but still just a fraction of the cost of the overall move), and were personably professional and cordially competent throughout.
Note that it will facilitate communication if, on your end, you have access to a scanner. That way, you can scan each and every document for remote review by their legal team, which saves considerable time over repeat iterations. (Fewer iterations if you actually follow their directions..."sign every page means sign every page!")
Of most interest will be Home Office processing times...unfortunately, those seem to vary considerably. I'm told they tend to be shorter in the winter, even as low as 3-4 weeks, while the "summer crunch" can see queues stretching to 12-14 weeks. Note that an HSMP, once approved, does not "start the clock" with regard to the 2yr work window until you actually set foot in-country, which I believe can be anytime up to 6mo after approval. Therefore, there is little risk in starting the process ahead of time, e.g. as soon as you have 12 months of appropriate documentation assembled.