Two different books are giving two different solutions.
The difference is that a $50$% loss and a $50$% gain (in either sequence) result in a net loss (AM-GM inequality), whereas halving and doubling (in either sequence) do not result in a net loss. Joshi is presenting (and solving) a different problem, one in which half the time the trader's expected return is $100$%. So there's no a priori reason to expect the same result.
Having said that, Wilmott's answer to the Joshi question is wrong. For $n$ tosses, $R_k=2^k(\frac 12)^{n-k}=2^{2k-n}$, where $k$ is the number of times you toss heads. Wilmott's analysis of Joshi assumes that you are starting afresh each time with a single dollar.
Wilmott's solution to his own problem is correct. If you take ten trials, you expect a return of $1.5^6 \cdot 0.5^4 -1 = \frac{729}{1024}-1 = -\frac{295}{1024}$. Taking the geometric mean gets you $\sqrt[10]{1.5^6 \cdot 0.5^4} -1 = 1.5^{0.6} \cdot 0.5^{0.4}-1$, which is exactly what Wilmott says (just writing it in exponential form).
They’re computing two entirely different things. Wilmott is computing the minimum number of days out of $260$ on which you must make a profit in order to come out ahead; Joshi is computing the expected value of your portfolio. Applying Joshi’s calculation to Wilmott’s setting, we get an expected value after $260$ days of
$$(0.6\cdot1.5+0.4\cdot0.5)^{260}=1.1^{260}\approx 57,833,669,934\;.$$
Wilmott’s calculation does not take the probabilities of the two outcomes into account: it would yield the same result whether you made a $50\%$ profit with probability $0.99$ or with probability $0.01$. In the former case, however, you are almost certain to make a net profit, while in the latter you are almost certain to lose virtually everything. No matter what the probabilities are, you need to make a profit on at least $165$ days in order to come out ahead for the year; your likelihood of actually doing so, however, changes greatly with the probabilities.
In the original problem you might find it odd that the expected number of days on which you make a profit is $60\%$ of $260$, or $156$ days, and you lose money if you make a profit on exactly $156$ days, yet your overall expected value is enormous. This is because once you reach the break-even point, your expected final value grows explosively as the number of profitable days (out of $260$) increases, and these huge profits more than compensate for the more likely losses.
If you want to know how likely it is that you’ll make a profit, you want Wilmott’s calculation; you can then plug the figure of $165$ days into a binomial distribution calculator and find that the probability of making a profit on at least $165$ days is only about $0.14$. The fact that the expected profit — expected in the mathematical sense, that is — is considerable would probably not be very comforting, since it results from the fact that relatively unlikely outcomes produce huge profits.
The crucial thing is that Wilmott asks about the chance of making a profit, regardless of how large the profit or loss is. Joshi is asking about expected value of the portfolio. Those are very different questions. If I pay $1$ to bet on something and win $10$ with probability $\frac 15$ but can only play once, Wilmott says I should not. I lose $80\%$ of the time. Joshi says I should play, because my expected return is $2$. They are asking different questions and getting different answers.