Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags...
I am a bit confused about the subject in the title. For simplicity and concreteness, lets take the canonical simplest example $G=SL_2(\mathbb{C})$, with Borel $B$ the upper triangular matrices, its unipotent radical $U\cong \mathbb{G}_a$ the strictly upper triangular matrices and the maximal torus $T$ of diagonal matrices (all of this of course in $SL_2$). Denote $X=G/B$ the flag variety.
The standard construction of a line bundle on $X$ proceedes as follows. Let $\lambda:T\to \mathbb{G}_m$ be the simple positive character of $T$, we can inflate it to $\hat{\lambda}:B\to \mathbb{G}_m$ and define the action of $B$ on $G\times \mathbb{C}$ by $b(g,x)=(gb^{-1},\hat{\lambda(b)x})$. Finally, dividing by the action of $B$, we get $\mathcal{L}=G\times ^{B}\mathbb{C}$ and the map to $X=G/B$ induced by projection to the first factor realizes $\mathcal{L}$ as a line bundle on $X$. Hope I am ok so far.
Now, another possibility comes to mind. We have a map of $G$-homogeneous spaces $G/T\to G/B$ whose fiber at the identitiy (and thus everywhere up to isomorphism) is $B/T\cong \mathbb{G}_a$. Moreover, the map has a local section (on the big cell). Hence, this is a fiber-bundle with fiber $\mathbb{A}^1$. But, is this a line bundle? i.e. are there trivializations with linear transition maps on the fibers? What are they? If it is, is it the same line bundle from the previous paragraph?
If I use the standard section on the big cell and it $G$-translates, the transition maps don't seem to be linear, but perhaps I just miss something obvious.