I do know that something can be injected into a patient's IV to wake them up immediately from general anaesthesia.
That's not how it works.
Many anaesthetics only work for minutes at best. A normal dosage of propofol would have to be renewed every 5-10 minutes. To "wake" you from general anaesthesia, they just stop administering anaesthetic agents.
The duration of action of intravenous induction agents is generally 5 to 10 minutes, after which spontaneous recovery of consciousness will occur. In order to prolong unconsciousness for the required duration (usually the duration of surgery), anaesthesia must be maintained. This is achieved by allowing the patient to breathe a carefully controlled mixture of oxygen, sometimes nitrous oxide, and a volatile anaesthetic agent, or by administering medication (usually propofol) through an intravenous catheter. Inhaled agents are frequently supplemented by intravenous anaesthetics, such as opioids (usually fentanyl or a fentanyl derivative) and sedatives (usually propofol or midazolam). With propofol-based anaesthetics, however, supplementation by inhalation agents is not required.
At the end of surgery, administration of anaesthetic agents is discontinued. Recovery of consciousness occurs when the concentration of anaesthetic in the brain drops below a certain level (usually within 1 to 30 minutes, depending on the duration of surgery).
It mostly depends on the skill of your anaesthetist how deep the general anaesthesia was, and when they stopped administering anaesthetics. This is mostly a timing issue, and has little to nothing to do with different nations. Regarding the "they were expecting me to wake up", there are certain hints: Patients start to move more, the gagging reflex kicks in, heart rate slowly rises, they react to pain and so on. With a bit of experience, the nurses can tell when you're about to wake up.
As to my knowledge, there is no medication that can reverse general anaesthesia, but they experiment with Ritalin in rats.
Regarding the duration of action for anaesthetics:
For barbiturates (midazolam, lorazepam, diazepam) it is 5-10 minutes:
the short-acting agents have a rapid distribution, and thus a short duration of action of 5–10 minutes following a single bolus dose. If a more prolonged sedative effect is needed, a continuous infusion of these agents is required to maintain plasma concentrations
For N2O. the duration of action is likewise low:
N2O has a rapid onset of action, is relatively easy to use and is inexpensive. It provides amnesia, sedation and analgesia. Its effects dissipate rapidly when administration is discontinued.
Propofol has a half life of 1-3 minutes:
As with other rapidly acting intravenous anesthetic agents, the half-time of the blood-brain equilibration is approximately 1 minute to 3 minutes, accounting for the rate of induction of anesthesia
Fentanyl and remifentanil also have a half life of around 9 minutes:
Remifentanil is an ultra-short-acting opioid that is rapidly metabolized in the plasma and tissue by nonspecific esterases to an inactive metabolite. It has a very brief elimination half-life with a context-sensitive half-life of only 9 minutes, independent of the duration of infusion.
Source: McEwan, Angus: A Practice of Anesthesia for Infants and Children (Fourth Edition). CHAPTER 15 - Anesthesia for Children Undergoing Heart Surgery. 2009