After rewatching the Star Wars saga recently I became curious as to when Palpatine was certain that he would pursue Anakin as his new apprentice.
TL;DR Somewhere between Episode I and Episode II, Palpatine committed to taking on Anakin as an apprentice.
As mentioned in the OP, Palpatine first became aware of Anakin at the end of Episode I, saying,
"And you, young Skywalker, we will watch your career with great interest."
Going into Episode II, there's an assassination attempt on Padme's life (plane exploding), and Palpatine used it as an excuse to suggest that Anakin be Padme's bodyguard.
PALPATINE: Master Jedi [Windu], may I suggest that the Senator [Amidala] be placed under the protection of your graces.
I realise all too well that additional security might be disruptive for you [Senator Amidala], but perhaps someone you are familiar with... an old friend like... Master Kenobi...
Do it for me, M'Lady, please. I will rest easier. We had a big scare today. The thought of losing you is unbearable.
By doing this, Palpatine was slowly coercing Anakin to the Dark Side by having him indulge in temptation/lust/fear/anger, which stemmed from his romantic relationship with Padme. And even from the very beginning of the protection assignment, Anakin was already on this path:
ANAKIN: She didn't even recognise me, JAR JAR. I thought about her every day since we parted... and she's forgotten me completely.
ANAKIN: I am grown up. You said it yourself.
ANAKIN looks deep into PADME'S eyes.
PADME: Please don't look at me like that.
ANAKIN: Why not?
PADME: Because I can see what you're thinking.
It makes me feel uncomfortable.
Once Anakin and Padme were on Naboo, alone, they inevitably fell in love; and, by the end of the movie they get married. This is exactly what Palpatine wanted.
Or was Palpatine resilient in his plans and use of Count Dooku only to call an audible once he became witness to Anakin’s true power as a Jedi?
Palpatine had no intention of keeping Dooku around.
Count Dooku's primary purpose [within the context of the OP] was to die at the hands of Skywalker, serving as Anakin's first cold-blood murder (an order that was directly issued by Palpatine, and, is the first time you see Palpatine behave as Darth Sidious).
DARTH SIDIOUS: His [Dooku's] death was a necessary loss, which will ensure our victory. Soon I will have a new apprentice ... one far younger and more powerful than Lord Tyranus.
After Anakin kills Dooku, there's a brief conversation in which Palpatine begins teaching Anakin the ways of the Dark Side (though it's subtle).
PALPATINE: You did well, Anakin. He was too dangerous to be kept alive.
ANAKIN: Yes, but he was an unarmed prisoner. I shouldn't have done that. It's not the Jedi way.
PALPATINE: It is only natural. He cut off your arm, and you wanted revenge.
PALPATINE: Remember what you told me about your mother and the Sand People.
This scene is truly when Anakin is first taken on as an apprentice by Darth Sidious, and, in a sense, that conversation is Darth Sidious' first lesson for Anakin, reinforced by Anakin's previous actions of slaughtering the entire village of sandpeople who killed his mom (Palpatine appealing to this logic & reasoning was critical in order for Anakin to accept that revenge is the "right" thing to do, in some sense).
And lastly, when they're leaving the throne room, Palpatine tries to get Anakin to leave Obi-wan behind. He knew that if Obi-wan was around, he'd have a much more difficult time converting Anakin to the Dark Side.
PALPATINE: Anakin, there is no time. We must get off the ship before it's too late.
Leave him, or we'll never make it.
ANAKIN: His fate will be the same as ours.
So, again, all of this started at the beginning of Episode II, and, since Palpatine had only just become aware of Anakin at the end of Episode I, his decision to focus his efforts on Anakin happened between those two movies.
Additional reasoning: One could easily argue that, should Palpatine had known about Anakin before the end of Episode I, he would have just sent Darth Maul (or someone else) to get Anakin & start training him immediately, before Anakin had joined the Jedi. Anakin was a slave after all.. Palpatine could have just bought him, or taken him forcefully.