Sum of higher powers (bound on $N$) Sums of cubes and more
In the selected answer to the above question the writer states ""what is the least $G=G(k)$ such that for some $N$, every integer greater than $N$ can be represented as the sum of $G$ $k$-th powers"
For $G(k)$ that we have explicit bounds in http://www.maths.bris.ac.uk/~matdw/2002%20wps.pdf.
Does anyone know any bounds for the values of $N$ for any $k$ would be?
 A: For the first few relevant cases even exact values are known or conjectured. Fairly recent work on this was done by Deshouillers and different groups of coauthors:
In particular, it is known that for $k=4$ that, recall $G(4)=16$, that $13792$ is the last number to requier $17$ fourth powers, and this is thus your (optimal) $N$ for $k=4$.
And, it is conjectured that for $k=3$ one has $G(3)=4$ and $7373170279850$ is the largest value requiring more than $4$ cubes and this number thus would be the (optimal) $N$ for $k=3$. Note however that only $G(3)\le 7$ is known.
Regarding the more general problem/your question some remark (which I hope to make sense):
As long as one does not know $G(k)$ it might be difficult (or impossible) to give bounds for the $N$ relative to $G(k)$.
Look at the problem like this: let $N(k, m)$ denote the supremum (so allowing infinity) of all $N$ that are not the sum of $m$ $k$-th powers. 
For fixed $k$ this $N(k, m)$  is a nonincreasing function in $m$ and determining $G(k)$ is precisely the problem of determining the smallest $m$ where it is finite.
Now, the bounds on $G(k)$ you mentioned give an upper bound on this $m$. 
I believe that the proofs used to establish these bounds, at least in principle, will also yield a bound on $N(k, m')$ where $m'$ is at least as large as the bound for $G(k)$. However, this is than not a bound for the $N(k, G(k))$, your $N$.
So, in summary, I find it unlikely that a so to say 'abstract' bound on $N$ can be established without knowing (or conjecturing) a precise value for $G(k)$;
By contrast, I believe that for $m'$ at least as large as a bound known for $G(k)$ one can extract an explict value for $N(k,m')$ from the proofs (in case this was not done by the authors themselves).
The Wikipedia page on Waring's problem contains precise references for the results mentioned above. 
