Thanks for your Question.
It is going to be difficult. If the children do not have any British Ancestry then you are looking at having to make an application for settlement for them as your dependents. Nieces are not considered dependents automatically in the same way that your children would be. This means that nieces are only accepted as dependents if there are exceptional compassionate circumstances that apply to them.
The burden is high. The key is proving actual dependency at the present time, which is that they are living out of the UK and surviving because you are mainly responsible for providing finance to them to enable them to live. If you are not doing this then the chances of a successful application are very remote.
If you are providing them with money to live then it must be a sufficient amount so that if it was not paid then their welfare would be severely affected. You would have to submit substantial documentary evidence supporting dependency (bank statements, evidence of the conditions they are living in, possibly doctor's notes etc) as well as detailed statements from all parties arguing that they cannot live with our the financial and familial support you provide. Jamaica is not considered a third world country either, so there is a further perception issue to overcome there.
In addition you would also have to show that you can accommodate them and support them without any needing to access any more public funds than you already do. Obviously you would also have to prove your relationship to them by submitting each party's birth certificates and passports.
If you were to make an application I would strongly advise you to get a solicitor to advise you initially on the specific merits of your application and, if you are advised that there is a reasonable chance of success, to draft the application on your behalf. It would cost around £1000.00+vat to prepare the application and if it is rejected you will have to appeal to the Immigration Appeals Tribunal. For this you would need the solicitor to act and to instruct counsel on the day of the Hearing. The costs for this appeal would probably be around £2000-3000.00.
I am sorry it could not be better news.