I delayed telling him about his uncle’s death as long as I could.
死のことを話す is very wordy and unnatural. 死のこと is a word-for-word translation from “about the death”. but we don’t say 死 + こと. We also usually use a verb 告げる (lit. = to inform) in the context of such a sad news in a respectful way. 話す is more like “to lively discuss” or “to have a chat”. For example, 告別式（こくべつしき）is “a funeral” (lit. = a ceremony to say goodbye to the deceased person).
先送りする also sounds unnatural a bit in this context although it’s understandable. 先送りする is more frequently used for “to postpone (a meeting)”, “to procrastinate (a task)” or “to sleep on something (important decision) for several days/months” – all of them are business-like tasks. But we don’t usually use 先送りする when other people “refrain” from telling the truth of someone’s death for a while. It sounds too dry with no condolences.
Here is my alternative translation:
FYI: an “uncle” can be spelled as either 伯父 or 叔父, both of which are pronounced as おじ. 伯父 is someone’s parent’s older brother whereas 叔父 is someone’s parent’s younger brother. The Kanji 伯 is a honorable prefix, meaning “a leader”, and Japanese people usually respect for elder people with a strong influence from Confucian. The English sentence doesn’t specify, so both 叔父 and 伯父 work as a proper translation.
If the speaker (I) and him are close enough, or if the situation is not so formal, you can conjugate 伯父/叔父 and さん.