I assume there is no will and that your uncle died intestate domiciled in England or Wales.
Assuming there are is no surviving spouse, children or parents, the next set of relatives to inherit are siblings. If the siblings are alive, they inherit a proportionate share. If the siblings have predeceased without issue, they do not take. If the siblings have died leaving issue, they take per stirpes (literally through the stems). If the siblings have died afterwards - that share is distributed according to that sibling's intestacy or will. Easiest way to explain is by example.
Arthur dies intestate. He had 4 brothers. Bert who predeceased without children. Colin who has survived him. David who died before Arthur but left two children, Fred and George. Ernest who has survived Arthur but died 2 months later leaving a will which left his whole estate to Charity X. Ernest also has two children Henry and Isaac. Who gets what?
Bert's "stem" has died out completely and is discounted. The estate is therefore divided into 3 shares. Colin receives one share. Fred and George receive one share split between them. Ernest's estate receives one share but Henry and Isaac get nothing and Ernest's share goes to Charity X through Ernest's estate. So yes, it sounds as if your cousins should inherit.
As for the debts going to the next of kin, strictly speaking they dont. The assets in the estate are gathered in and the debts settled out of the assets. Once all the debts, testamentary expenses and tax is deducted, what is left is divided as above.