Q: Does the buyer's bank have an obligation to complete their actions on a home sale in a certain period of time?

A: You have no privity with the buyer's bank, and they have no duty to you. Indeed, they have a duty of confidentiality with their customer, such that they should not tell you if the real problem here is the buyer. Your contract with the buyer controls your relationship. The buyer's contract with the bank controls theirs.

A: Your remedies are strictly with your buyer, pursuant to the terms of your real estate contract, and not against the buyer’s bank. Breach of contract remedies for a seller are either declare the contract breached and forfeit the deposit, or sue for specific performance. Either way, your seller of the new house will likely not stick around waiting for you to relist your house or force the buyer to purchase yours without the financial means to do so. When accepting a contract to sell your home, it is not always the contract with the highest offer that should be accepted, but the contract with the buyer with the strongest financials that is most likely to qualify for the mortgage that is the best contract to accept. This will s especially true when you have made your ability to buy a new home contingent on selling your old home. Now, both contingencies (yours and your buyer’s) may allow both of you out of your contracts with no financial loss of your deposits and no obligation to buy. Not what you want, but this is why you pick and choose your buyers as carefully as you select the new home you want to buy.