What I went through
I've been researching this for a number of months ahead of my move and didn't come up with much info, until I actually got to the UK and started 'shopping around'.
Tried and failed..
I actually tried the 'Engineered Way' mentioned here, however neither of the two banks would accept it as valid, as their requirements clearly stated, they need both a Sort Code as well as the Account number. The Sort Code is the equivalent to the Routing Number (in the US at least). My statement (from a US bank) only included an Account number. Thus it wasn't a valid bank statement they would accept as valid proof of address. Their requirements are they need a Bank Statement, but explicitly from the UK to be precise (by virtue of needing it to have the Sort code).
The holy grail..
Turns out it's a lot more straightforward than I initially realized (and if I knew about the following sooner, I would have opened my account quicker rather than wait almost a month for the engineered way to be delivered, tried and tested).
This should work for you if you are employed by one of the companies that is recognized by uk.gov as having the ability to sponsor employee's. Most companies are recognized if they are major enough, and there are several tens of thousands of companies on the list. Here's the full list:
https://www.gov.uk/government/publications/register-of-licensed-sponsors-workers
If you find your company on the list (it's a huge list so take your time searching through it), all you need is to ask your HR department to write you a letter stating your info. Their word as proof is enough for the major banks (at least for the two I tried near central London). Of course you also need your passport for proof of identity, but this should be the least of your troubles if you managed to come to the UK :-)
The exact info wanted varies by bank but to be as complete as possible, you'll want:
- Full (Legal) Name
- Address currently residing
- Previous address (even if abroad)
- When you started working for the company
- Your current salary
My experiences with two major banks..
Barclays for example didn't need all this, HSBC did.
Barclays was content with just the first two items in a simple letter format (and possibly the 4th, as it was part of my letter but the 3rd and 5th item weren't, and i got no push back). The branch I went to was 114 Fenchurch and I eventually opened my account by "David" there (don't recall his last name). They were very helpful (albeit David was a bit serious, didn't seem friendly initially, and not until the end did i even see him crack a smile, i blame it on the "British reserve").
Contrary to my solemn (but productive) experience at Barclays, HSBC were at least super friendly (and the lady who helped me was quite attractive), but weren't very helpful in being to accomplish much initially, and I had to go back several times to get them all the info to them in the exact way they wanted. The branch I went to was at 60 Fenchurch (across the street and down a block from Barclays).
HSBC didn't like the initial letter I brought them, and even gave me a template of what the letter ideally would look like for them: 
Will my account exist long term?
Possibly as a result of being also a US citizen, at the end, they asked me for my 'National Security Number', to which i responded with a blank stare. They weren't referring to my National Insurance (NI) number as it was related to foreign jurisdiction they told me. They said it's a new policy, because of some agreement with the US and the UK. Barclays didn't ask me for this at all (maybe just because Barclays doesn't do retail banking services in the US).
I asked them if they actually meant a social security number? They would accept that but weren't sure if that's what they needed (and couldn't really elaborate on what this number is for).
I eventually declined to give it to them (the US Gov't already encroaches too much on people's rights stateside, I don't want them poking their nose in my business abroad also), and they reluctantly accepted my refusal but told me that if their back end folks aren't happy with that, they may contact me about closing my account (how's that for making you feel welcome). I really am not impressed with HSBC (and I told them that), and as long as I can keep my credit card account (to build credit) I won't miss HSBC. Maybe i'll use their statement to open/switch to another bank (see below).
I need an appointment ??
HSBC (and Barclays at one point) prefer if you make an appointment for another day to sit with the bank representative to open an account (HSBC insisted on it because they invasively ask you a bunch of questions about your life and financial situation, Barclays recommended me to make an appt just so i don't have to wait but weren't as nosy.. i was able to see someone by waiting a little bit). If you've banked in the US, this is somewhat unheard of. HSBC really made me feel like i was just a number to them with all this attempting to sell extra services (despite them trying to make up for it with friendliness), Barclays was a lot more accommodating and straight to the point, and while I initially had reservations about them, they were the better experience. It was Barclays afterall that told me about the list of companies that if i work for one of them opening an account would be a cinch, HSBC made zero mention of this.
Why HSBC?
My initial preference was with HSBC, because I already have an HSBC account in the states. They claim they can import your credit history. This was totally misleading in that, when i see the word 'import' i think of certain things, yet here there is no actual making it available in the UK (like 'import' i thought would imply). The US credit history (and this by the way, is only limited to your US HSBC credit account not your other institutions' credit histories) is merely referenced one time, only by HSBC UK for opening an HSBC credit account.
The problem for me was, I initially thought this was the answer to all my concerns about having a blank slate of a credit history in the UK, that when I went to rent a flat, it would show at least some useful, re-assuring info to the UK rental property managers doing a referencing check on me (and the size of the deposit they ask you for, is based on this, i was informed). Unfortunately this is not the case at all, and the wording on their web site is rather misleading. As such, i may now have to put down 2-3 months rent as a deposit (when normally only 4-6 weeks is asked for if you have good credit, in the UK).
Another service of HSBC i investigated was having them actually open the bank account for you before you arrive in the UK. This service is free for Premier accounts, $100 for Advanced accounts, $200 otherwise. This turns out takes 4-6 weeks, although when i first called several months before the move, they initially told me 10 days, so i called two weeks before my departure and was pretty disappointed.
Come to think of it, disappointment and HSBC go hand in hand, but that's another rant (their online banking looks like it was invented in the 90's and never quite got updated to modern web technology).
The Fee-Free catch..
My other main interest in HSBC was that there's 'free' (aka no Fees) for transferring money between my US bank and my UK bank (both HSBC), using their Global Funds Global Transfer service. This again is a misnomer, as they inflate the exchange rate 3% (so rather than charge you a flat fee, they just skew the rate they give you is 3% off the market rate in their favor, perhaps this is just what everyone does, not sure yet). Oh and they told me that it would be free outright (in the UK). Sure enough, when i tried it, from the US to UK didn't have any extra fees (except the crappy exchange rate), BUT when i tried from UK to US, it states I'd be charged 5 GBP on top (and that fee this time is skewed 4%). Now i don't know if that's because i have HSBC Advanced in the US and not in the UK (and HSBC Advanced isn't available in the UK at the time of this writing, i'm told this will change next month (Nov 2014)), but being let down by HSBC is a constant theme.
What's next?
My next bank I will visit is Citibank and they also told me about the fee-free international transfer between Citi accounts (and i have a US one already so just need to see if the UK one will live up to the hyper). Will see if this fee-skew happens there too.
Also, i highly recommend spending some time on this site, as there's a load of financial related advice and information, that should help you with your financial life in the UK. Here's the section on credit cards:
http://www.moneysavingexpert.com/credit-cards/
But they cover other topics such as bank accounts, ISA's, etc.
Anyway hope this helps other people, so they don't have to go through the struggles I and many others have had to go through.