Since Life Insurance Payments are almost always guaranteed especially with a major life insurance company then yes the Funeral Home will wait for the payment from the Life Insurance Company to pay out and will proceed with the funeral.
You need to provide the Funeral Home with documents of the life insurance policy and they will make contact with the life insurance company to have them payout the life insurance policy to cover the funeral costs.
You should also make contact with the life insurance company so you know what's going on.
As long as the Life Insurance policy is in effect and has not been cancelled then the life Insurance should pay out on the death of the person and pay for their funeral and any legit funeral home should wait on the life insurance policy.