The question is reasonable (even in the full generality of finite groups of Lie type in the defining characteristic $p$). However, the answer requires case-by-case study, as in related questions about classifying unipotent classes and centralizers of unipotent elements. Most of the information needed starts out in the ambient algebraic group $G$ and leads to fairly uniform results for the finite groups when the prime is "good" (meaning odd for symplectic or orthogonal groups). When studying unipotents, the isogeny class of the group doesn't matter, or whether you consider general linear or special linear groups. In good characteristic it's mostly enought to look at nilotents in the Lie algebra of $G$, but bad primes require some extra study within the group setting.

Probably the most straightforward way to think about the exponent of a Sylow $p$-subgroup (coming from a maximal unipotent subgroup of the algebraic group) is to look at a smallest possible faithful representation of $G$, say of dimension $n$. Since the $n$th power of each nilpotent matrix vanishes, a rough upper bound on the possible orders of unipotents in $G$ is given by the *least* (not largest) power of $p$ which is $\geq n$.

What about a precise determination of the exponent? Since there are only finitely many unipotent classes in $G$ (even for bad primes), you mainly need to locate the largest Jordan block size among those occurring for any unipotent matrix in a class. If this size is $d$, then the exponent is the least power of $p$ which is $\geq d$. For classical groups, at least when $p$ is odd, the unipotent classes in $G$ and then a finite subgroup of Lie type are fairly easy to obtain from those of the ambient general linear group. There is a lot of published work in this direction, going back to a paper by G.E. Wall. Useful books include the 1985 *Simple Groups of Lie Type* by Roger Carter and a recent AMS book *Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras* by Martin Liebeck and Gary Seitz.

Probably the *regular* unipotent elements have the largest orders, though in the symplectic and orthogonal groups they need not involve just a single large Jordan block as they would in the special linear group. Anyway, each type and each small prime need some attention here. For finite or algebraic groups of *exceptional* types, all of this is more delicate in terms of working with a convenient matrix representation (the adjoint representation being the smallest for type $E_8$). So much attention in the literature is directed at these groups: see for example the influential paper by Ross Lawther: *Jordan block sizes of unipotent elements in exceptional algebraic groups*,
Comm. Algebra 23 (1995), no. 11, 4125–4156.

Final remark: In the study of finite simple groups, the exponent is not usually a major player. For example, it doesn't show up directly in the character table. The main themes have been the classification of conjugacy classes and the determination of corresponding centralizers in $G$ and the finite groups.

ADDED (post-final remark): Looking more closely at known results, I can see that the exponent varies somewhat erratically over powers of $p$ until it seems to stabilize at $p$ when $p \geq h$. Here $h$ is the *Coxeter number* (not the rank), which in types $A_n, B_n, C_n, D_n$ is respectively $n, 2n, 2n, 2(n-1)$. If this bound on $p$ is optimal, it's reasonable to look for a general proof. It seems consistent with Lawther's tables for exceptional types. (It's also worth comparing with work on elements of prime order in simple complex or compact Lie groups.)

P.S. As Maria points out, there is an exact formulation for all Lie types and all primes in Corollary 0.5 of the 1995 paper by Testerman. I had forgotten about this, but her work and Lawther's have been closely related, and sometimes collaborative as in their two AMS Memoirs no. 674 (1999) and no. 988 (2011). As far as I know, the proof of 0.5 near the end of her paper requires considerable case-by-case work (most of which assumes $p$ is good) together with separate treatment of bad primes. By now there is a lot of related literature, including the recent monograph by Liebeck and Seitz, so I'm not quite sure what the best arguments are. (Also, Maria has corrected my hasty listing of Coxeter numbers: $\mathrm{SL}_{n+1}$ has type $A_n$ and Coxeter number $n+1$.)