That regular expression represents a Markov chain on $m+1$ states corresponding to a start state $s$ and each of the letters. A transition is made from $s$ to $a$, from $a$ to $b$, ..., and from the penultimate letter to the last, always with probability $p$. Otherwise the state remains the same. The final state is an absorbing state: when it has been reached, all letters have been observed in sequence.
In terms of the states $(s, a, b, \ldots)$, the transition matrix is
$$\mathbb{P}_m = \pmatrix{1-p & p & 0 &\cdots & 0\\
0 & 1-p & p &\cdots & 0 \\
\vdots & 0 & \ddots & p & \vdots \\
0 & \cdots & 0 &1-p & p\\
0 & 0 & \cdots & 0 & 1
}$$
Standard linear algebraic techniques (the Jordan normal form of $\mathbb{P}_m$ and its change of basis matrix are simple and sparse, making this fairly easy to do) establish that for $n\ge m$ the last entry in the first row of the matrix power $\mathbb{P}_m^n$ is
$$\mathbb{P}_m^n(1,m+1) = p^m \sum_{i=0}^{n-m} \binom{m-1+i}{m-1}(1-p)^i.$$
This is the chance of reaching the absorbing state from the start state after $n$ transitions: it answers the question. If you like, it can be expressed in "closed form" in terms of a Hypergeometric function as
$$\mathbb{P}_m^n(1,m+1) =1-p^m \binom{n}{m-1} (1-p)^{-m+n+1} \, _2F_1(1,n+1;n+2-m;1-p).$$
The sum has a pleasant combinatorial interpretation. Let $m+i$ be the position at which the last letter first occurs. It is preceded by a (possibly empty) sequence of non-$a$s, each with a $1-p$ chance of occurring; then an $a$ with a $p$ chance of occurring; then a (possibly non-empty) sequence of non-$b$s, etc. There are $\binom{m-1+i}{m-1}$ locations at which to place the first appearance of an $a$, then the first appearance of a $b$ after that, etc. Thus--including the first appearance of the last letter in position $m+i$--the probability is $\binom{m-1+i}{m-1}p^m(1-p)^k$. This gives one term of the sum. Thus, the sum breaks up the sequences according to where the last letter first occurs, which can be anywhere from position $m+0$ through $m+(n-m)$--these are obviously disjoint--and adds up their probabilities.
As a simple example to clarify the interpretation, suppose $m=2$ and consider $n=3$. There are four sequences of three symbols, each of probability $p^3$, and three other sequences of probability $p^2(1-2p)$, in which the symbols $a$ and $b$ appear in order:
$$aab, aba, abb, bab; ab\$,a\$b, $ab.$$
The chance therefore is
$$4p^3 + 3p^2(1-2p) = 3p^2 - 2p^3 = p^2(3-2p) = p^2(1 + 2(1-p)) = \mathbb{P}_2^3(1,3).$$
The combinatorial interpretation is that the regular expression ^ab (with $b$ in position $2$) occurs with probability $p^2$; and ^[^a]*a[^b]*b, with $b$ in position $3$, occurs in two ways as ^a[^b]b and ^[^a]ab, each with probability $p^2(1-p)$.
