Is it permissible to carry out the will of a deceased sister to her niece for all of her gold, with the knowledge and consent of her husband? And would this entail sin or family sensitivities?

It is permissible for a person to make a bequest to a non-heir, provided it does not exceed one-third of the estate, based on the Prophet's (peace and blessings be upon him) saying: "Indeed, Allah has bestowed upon you as charity, at the time of your death, one-third of your wealth, as an increase for you in your deeds."

Therefore, there is no harm in your sister's bequest of gold to your daughter, because it is a bequest to a non-heir. If the gold does not exceed one-third of her estate, then the bequest is valid and enforceable, even if the heirs do not consent. If it exceeds one-third, then your daughter is entitled to only the amount equivalent to one-third, and anything beyond that is not enforceable except with the consent of the heirs.