A financial crisis can have a lot of impact on lending, in many different ways. Right now, It's not super clear what your understanding of lending (in general) is, or exactly what factors you're interested in. Talking about a bank's "willingness" to lend is somewhat abstract. Banks are generally always in the business of being willing to lend money, but how and when and under what terms may change over time. It might be more appropriate to focus on how a bank makes decisions about the following:
- loan features (interest rate, term, etc). Interest rates are based on risk. If a bank expects a lot of defaults, they will typically raise rates.
- which products to offer (should a bank offer adjustable rate mortgages, or just fixed rate?) Banks may aggressively promote certain products based on what triggered the crisis.
- who to lend to (obvious things like, only lend to A or B paper, but also less obvious things like, don't write HELOCs or home equity loans in second position behind someone else's loan, only write them behind our own loans). During or immediately after a crisis, a bank may be less likely to write a loan behind another lender.
- how to manage the lending portfolio after the loans are written (should we keep these loans, or sell them? Which loans should we sell, under what terms, to whom? When we sell loans, do we continue to operate them, and do we keep any of the risk?) Portfolio management is an important variable for banks when they expect the financial environment to change. Depending on the trigger, they may decide to buy or sell more of a specific type of loan, or loans to specific types of consumers.
Timing of a crisis is important in these decisions. If a lender suspects a crisis is coming, they may make one set of decisions. But, in the middle of, or after, an obvious crisis, they may make different decisions. Unfortunately, crises really only happen when not everyone expects them, so it's hard to pick out a generic, coherent answer to your question if you're thinking about the "ahead of time" aspect.
There's also a big factor in terms of regulation. To a large extent, for real estate loans, lending is regulated to the point that the federal government is impacting who gets loans, and how much banks are able to lend. So, even if a bank "wants" to write loans, the government plays an important role in deciding if they are able to or not. Typically, in the years after a crisis, a slew of regulations will emerge based on trying to target what the perceived cause of the crisis was.
And of course, there's also consumer behavior. People will behave differently depending on the crisis. Banks may be desperately trying to lend to anyone who walks in the door, but if consumers aren't interested in a certain loan, none of that matters.