Did Gandalf make a mistake at Isengard ?

[The Mouth of Sauron]

When Gandalf arrived at Isengard, much to the surprise of Merry and Pippin, he alerted Treebeard to the possibility that Wormtongue would be there soon.

And sure enough, Wormtongue turned up, was apprehended by Treebeard and was sent into Orthanc.

But was that not a tactical mistake by Gandalf ? Wormtongue SAW Merry and Pippin. If he had alerted Saruman to their presence at the ruined gates, Saruman could then have informed Sauron via the Palantir and a Nazgul could have been sent to Isengard to pick up the 2 hobbits within hours.

[Inziladun]

When Gandalf first arrived at Isengard to survey the scene and get news from Treebeard, he did not, in fact, know that Saruman was in possession of a palantír yet. For that reason, I can't fault Gandalf for not ordering Treebeard to restrain Wormtongue upon his arrival and not allow him into Orthanc.
I can see the potential for harm there, but even if Saruman had tried to do what you suggest, I don't think Merry and Pippin would have been in any real danger at that point.

'Oh yes, I forgot about (Wormtongue),'said Pippin. He did not get here till this morning.'

TT Flotsam and Jetsam

Even if Wormtongue had told Saruman of the hobbits' presence, as he no doubt did, and Saruman immediately contacted Sauron, I don't know if even a Nazgûl would have had time to get to Isengard before the arrival of Gandalf et al.
Also, would the Ents have allowed a single Nazgûl to make off with Merry and Pippin?

[Boromir88]

I don't even think Saruman could have contacted Sauron, he got stuck in a no win situation. The entire time Saruman was looking out for himself and planning to betray Sauron. Sauron eventually figured this out, but the point is, as Gandalf tells Theoden et all after Pippin looked in the palantir, Saruman got his caught in a vice, and his charade was up.

The "good" side knew Saruman was a liar, Sauron became aware of his intentions, and we see when Pippin uses the palantir Sauron assumes he is Saruman's servant and asks Pippin why his master had neglected to report in a while. Saruman had no escape, Isengard was destroyed, his power in the Council and in the Order of the Istari was gone, he was trapped in his tower, with no power to hand Merry and Pippin into Sauron whether he told Sauron or not.

Gandalf speculates that Saruman does have the power, while in Orthanc, to maybe trap one of the Nazgul, but other than that he's got nothing. Saruman's choices were be imprisoned in Orthanc and take a chance with the good guys who would spare his life, or let Sauron know that he was virtually powerless. Isengard was lost, even if he did tell Sauron he had two hobbits, Saruman was imprisoned in a tower, he couldn't hand them over to Sauron, and facing Sauron (even when he did have his power!) was the last thing Saruman wanted. Saruman was obviously bitter and spat right back in Gandalf's face, but they weren't going to kill him, it was better to be imprisoned by Gandalf et the "softies," then to let Sauron know he was powerless and had nothing to offer.

[The Might]

Boromir makes a good point of why he probably wouldn't have done in the first place, but as already stated even if he did inform him, there is no way a Nazgûl could do anything about it.
Proof?
The Nazgûl who flew over the camp on Dol Baran was not able do to anything. Now, indeed it did not directly respond to Pippin talking to Sauron through the palantir, the journey was too long and he had appeared immediately. He was actually sent to check on Saruman, but, even as its master detected Pippin and it was right above him it could not do anything about it.
So I doubt it could had done anything else sometime earlier.

[Ibrîniðilpathânezel]

Inziladun makes a good point, that Gandalf was not aware of Saruman's possession of a palantir until after it had been flung at him (contrary to what was shown in Jackson's version of the story). Moreover, when Pippin looked into the palantir, Sauron's first response was to think that it was Saruman reporting, which, for some reason, he had not done in some time. "So you have come back. Why have you neglected to report for so long?" From that reaction, I would hazard to guess that Saruman was no longer inclined to tell Sauron anything, unless it could save his skin, and especially not if he felt he might be able to use the information first, to his own advantage.

[Formendacil]

Originally Posted by Ibrîniðilpathânezel

Inziladun makes a good point, that Gandalf was not aware of Saruman's possession of a palantir until after it had been flung at him (contrary to what was shown in Jackson's version of the story).

Although the basic point here that Inziladun and Ibrin are making is to the effect that the palantír did not play a deliberative role in Gandalf's actions, I think it's incorrect to say that he did not know a palantír was in Saruman's possession.

The Orthanc-stone, after all, had always been at Isengard. Unlike the stones of Arnor, it was never moved due to enemy advances, and unlike Osgiliath and Minas Ithil, Orthanc was never taken by an enemy army--though, if my memory serves (and it IS hazy), the garrison did become strongly Dunlendish at some point, and resist Rohirric influence.

However, assuming that my memory IS right, then it also says that this was a huge part of the deliberative reasoning in Saruman receiving guardianship of Orthanc from the Stewards--Gondor wanted to know the palantír was safe in the hands of a wizard.

Granted, I doubt that the palantír was present much in Gandalf's thoughts--insofar as he knew, they weren't being used. It is hugely clear that the revelation of Gríma chucking the stone at him was of something he'd not known before... BUT... in the interests of pedantry, it seems fair to suggest Gandalf should have known it was there, even if he hadn't made the connection.