Here's a possibly helpful formula (that seems very hard to put in closed form). If you have already drawn $n$ cards, then $P(\mathrm{success})$ is
$ \sum_{\spadesuit+\heartsuit+\diamondsuit+\clubsuit=n;\ 0\leq \spadesuit,\heartsuit,\diamondsuit,\clubsuit\leq13}\frac{13-\min(\spadesuit,\heartsuit,\diamondsuit,\clubsuit)}{52-n}\cdot\frac{\binom{n}{\spadesuit, \heartsuit, \diamondsuit, \clubsuit}\binom{52-n}{13-\spadesuit, 13-\heartsuit, 13-\diamondsuit, 13-\clubsuit}(13!)^4}{52!} $
(This formula uses the multinomial coefficient, which in this case counts the number of ways to choose $\spadesuit$ positions to be spades, $\heartsuit$ of the remaining positions to be hearts, etc. out of the $n$ first cards to be drawn.)
The first factor is the probability of success given a partition $\spadesuit+\heartsuit+\diamondsuit+\clubsuit=n$ (the partition represents how the first $n$ cards were distributed, disregarding the order they came in). The rest is the probability of the first $n$ cards providing that partition (counting all the possible orderings, divided by the total possible orderings of a deck).
Added later:
The sum above conditions over the partition that represents the cards which have already been drawn. We could also sum over partitions that represent the cards yet to be drawn. That makes computing more efficient for the second half of the deck. If $r$ cards remain, then $P(\mathrm{success})$ is
$ \sum_{\spadesuit+\heartsuit+\diamondsuit+\clubsuit=r;\ 0\leq \spadesuit,\heartsuit,\diamondsuit,\clubsuit\leq13}\frac{\max(\spadesuit,\heartsuit,\diamondsuit,\clubsuit)}{r}\cdot\frac{\binom{r}{\spadesuit, \heartsuit, \diamondsuit, \clubsuit}\binom{52-r}{13-\spadesuit, 13-\heartsuit, 13-\diamondsuit, 13-\clubsuit}(13!)^4}{52!} $
Evaluating these sums requires enumerating partitions. This could be done with some programming, but I did it for $r,n\leq10$ in Excel and found the following $P_n(\mathrm{success})$:
$ \begin{align} P_0&=0.25 & P_1&\approx0.2549 & P_2&=0.26 & P_3&\approx0.2653 \\ P_4&\approx0.2686 & P_5&\approx0.2710 & P_6&\approx0.2733 & P_7&\approx0.2762 \\ P_8&\approx0.2788 & P_9&\approx0.2809 & P_{10}&\approx0.2830 & &\cdots\\ &\cdots & &\cdots & P_{42}&\approx0.4005 & P_{43}&\approx0.4122 \\ P_{44}&\approx0.4258 & P_{45}&\approx0.4392 & P_{46}&\approx0.4548 & P_{47}&\approx0.4836 \\ P_{48}&\approx0.5201 & P_{49}&\approx0.5514 & P_{50}&\approx0.6176 & P_{51}&=1 \end{align} $