Well, we moved here (from MSP actually, though I am not a Minnesotan) with no jobs and no credit and eventually (after we got ourselves employed) landed a decent enough 2 bed in outer zone 2 (West Hampstead) with an independent landlord. I had braced myself for dealing with the agencies and was thrilled that this actually came through. She was thrilled because it meant she didn't have to pay agency fees.
Because you are on a work contract then if you go the agency route they will probably still run the credit check (and charge you for it) but be ok with the fact that you have a work contract in hand. If you have the time and patience to look for an independent landlord I would recommend it, using gumtree.com (does anyone still use Loot? Even online?). For some bizarre reason the whole housing market persists with the foolish middleman agency system which really only serves to drive up prices further both on the weekly rate and the paperwork "filing" charges. If you are the primary breadwinner then I would base any personal budgeting on your salary and whatever your husband brings in as extra as it is so variable.
If you go independent - depends on the owner. Some want to see work info, some dont care. Some want to see bank info, some dont care. Ours took our word that we could afford the rent and we were employed, though it probably helped that we dressed nice and were already in a shared situation in the neighborhood.
It can be costly to move, so if you can find a place once and stay there, that would be best. The cost of housing is high, yes, but really the lack of choice, or even decent choice for the cost is the bigger challenge. You may have to make some concessions regarding quality and what you want and what you can afford.
Also, be sure to add an extra £200 a month to any rent rates to account for council tax and bills. Some places include these, most dont.
Finally, both of us work in the City and have fairly non-brutal commutes, so as long as you aren't moving to, say, Battersea or Clapham or anywhere connecting through London Bridge you should be fine. I take the train and its 20 minutes each way while the boyfriend takes the Tube and its about the same for him. We could maybe go two more stations further out into zone 3 and it would add an extra ten minutes, but its really because of weekend activities that we stay within the first two zones. I hated driving every day the same amount of time in the US, so find this much more tolerable!