Under the law, there is generally a presumption that both married parents hold joint parental responsibility over the children.
Since you are separated, you are still considered as lawfully married and therefore your husband still holds joint parental responsibility. This is why he would need to consent to your children returning with you to the UK.
If he does agree, you would need to put this in writing and preferably sworn before a solicitor or commissioner for oaths.
In the event that your husband refuses to give his consent, you should be able to get permission from the courts in Barbados to return to the UK with your children. In the event you don't, you risk that your husband files a complaint of child abduction against you.
Your application would be for leave of the court to remove children from the jurisdiction and would need to be lodged before the Family Division of the High Court. The Court is located at the Hall of Justice, Whitepark Road in Bridgetown.
In case you need a lawyer in Barbados, you can try one of law firms listed on the HG legal directory:
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