"I am particularly concerned that no mental health assessment was undertaken," the prime minister's statement said.
Death penalty defended
Shaikh was arrested in 2007 after he was found carrying a suitcase containing 4kg of heroin.
"My knowledge is that no one has seen him and this is despite the fact that in China they do have a flourishing and very professional psychiatric profession"
UK psychologist who was denied access to Shaikh
His family have said that he was tricked into carrying the suitcase by a gang in Poland, where he had been living for at least six years.
Melissa Chan, Al Jazeera's Beijing correspondent, said that China had pursued its historically tough line on drug crime.
"He [Shaikh] was caught with about eighty times the amount of minimum heroin required for a death penalty in this country.
"From the Chinese point of view, that is what they're focusing on, not the possible mental condition of the defendant."
A report on the court ruling by Xinhua, the Chinese official news agency, said that "the criminal regulations of our nation are implemented for all crimes equally and in accordance with appropriate laws".
"Therefore, crimes are dealt with equally before the law no matter what nationality." it said.
Al Jazeera's Chan said that Chinese law allowed for mental disorders to be taken into consideration.
"If they are found to have mental problems it is taken into consideration when it comes to the crime and to the sentencing," she said.
Seema Khan, Shaikh's cousin, told Al Jazeera: "He [Shaikh] has been assessed in the UK as suffering from bipolar disorder. And the evidence shows that he is displaying the symptoms of this disease."
But China's supreme court ruled that there was "insufficient" proof that Shaikh was mentally ill, Xinhua reported.
Dr Peter Schaapveld, a clinical psychologist, said he had travelled to China to try to assess Shakih's mental health but had been denied access to him by Chinese officials.
"They had ample evidence from myself, from Reprieve - the legal charity - and [from] the foreign and commonwealth office [the British foreign ministry] in the UK, that Mr Shaikh was suffering from a mental illness and had been for sometime," he said.
Schaapveld said he was "simply refused" access to Shaikh.
"I was given what looked like a couple of what appeared like excuses on the spur of the moment," he said.
"My knowledge is that no one has seen him and this is despite the fact that in China they do have a flourishing and very professional psychiatric profession."
Roseann Rife, the deputy programme director for the Asia Pacific at rights group Amnesty International, which has campaigned against the death penalty, said China needed to release details of its legal proceedings in such cases.
|Shaikh was arrested in 2007 after he was found to be carrying four kilos of heroin [AFP]
"The danger here is that we don't know exactly what evidence the supreme court looked at in coming to this conclusion [to execute Shaikh]," she told Al Jazeera.
"Because the application of the death penalty is considered a state secret in China, all of this remains behind closed doors. There's no transparency."
She also called for China to release figures on the number of people it executes.
Shakih had reportedly been unaware that his execution date had been set until he was told by his cousins, who visited him in Urumqi, the capital of China's western Xinjiang province, on Monday.
Shaikh, a father of three, who was born in Pakistan but was raised in the UK, is the first European Union citizen to be put to death in China since 1951.