Constant time range sum query
If the timestamps can take $T$ different values, then it is simple to implement the new, update, and delete operations in time $\Theta(T)$ and the query in time $O(1)$, by maintaining two arrays $val[ ]$ and $q[]$ of size $T$. The array $q[]$ is initially filled with zero values. 


new(x, v): set $val[x] = v$ and increment each $q[y]$ with $y\ge x$ by $v$; 
update(x, v): let $v' = val[x]$; increment each $q[y]$ with $y \ge x$ by $v-v'$, then set $val[x]=v$;
delete(x, v): let $v' = val[x]$; decrement each $q[y]$ with $y \ge x$ by $v'$;
query(x): return $q[x]$.


Note that $n$ (number of data items, that is, the number of total calls to new) is unrelated to $T$ (range of the timestamps). Whenever $T=o(n)$, the above implementation satisfies your requirements. For example, if a timestamp is a day of the year, $T$ will be a constant, even though $n$ can be arbitrarily large (because you can have arbitrarily many new and deletes). 

If, on the other hand, $T$ can be larger than $n$, then as D.W. noted in his comment your requirements may be too strict.