Underwriting standards have been tightened, and you won't pass screening.
Are you talking about a co-buyer, a co-borrower who has good credit rating? What about income?
Rule of Thumb: You may qualify to receive a mortgage to buy a home costing 2.5 times annual income, NOT more than 3 times annual income. Mortgage approval depends on the WHOLE package of you: credit rating, stable employment, sufficient income, substantial down payment, low debt load.
Lack of a bank account makes lenders look at you more carefully/suspiciously as it is unusual behavior in some one financially capable of buying and maintaining a home. And are you crazy, keeping this "substantial" down payment in your mattress rather than a bank? 10-20% is a standard down payment for conventional financing, not unusual. I would consider 35-50% down a substantial down payment.
BTW: Large cash (and credit) transactions are tracked, reported to IRS. That's what brought down Eliot Spitzer in NYC.
IMHO: You should open a bank account with your "substantial" sum of money, and start repairing your credit. In 6-12 months you should have improved it enough that you can buy a house unless you just went bankrupt or had a foreclosure. Pay off all delinquencies, pay down all outstanding debts, pay off all credit cards, pay everything in full and on time. Take a personal finance course, buy or listen to Dave Ramsay, take a first time home buyer's seminar. You will have improved your credit rating and self respect and be ready to become a home owner.