I've used a VA loan for a mortgage in the past. Typically, the process can take anywhere from 30 to 45 days, though it might vary based on specific circumstances and the lender you go with.
As for costs, you're looking at the VA funding fee, which can be rolled into the loan, and then the usual closing costs – appraisal, inspection, etc. You might want to shop around for the best rates and reviews.
When I was transitioning between properties, I worked with the
bankruptcy law center in San Diego. They helped me navigate through the situation. They might have a solution for you also.