Three-time Tour de France champion Chris Froome and five-time Olympic gold medallist Sir Bradley Wiggins were among those whose stolen files werepublished by Fancy Bears.
Ukad chief executive Nicole Sapstead said it is trying to contact athletes.
The files were stolen from the World Anti-Doping Agency (Wada).
They relate to therapeutic use exemptions (TUEs), which allow banned substances to be used for medical reasons.
Asked by BBC Sport how many British athletes could have their medical details disclosed, Sapstead added: "That's something we are determining at the moment.
"We are cross-referencing Team GB with the list of TUEs that are either currently in place or which have been historically granted.
"Those are the athletes we will be contacting in the first instance."
Sapstead said Ukad is also trying to make contact with the five athletes - Froome, Wiggins, golfer Charley Hull, rugby sevens player Heather Fisher and rower Sam Townsend, whose names have already been published.
She hoped speaking to them would give them some comfort and said Ukad would aim to give them "some re-assurance that we are working really closely with Wada about these leaks and trying to identify the specifics of exactly what has got into this cyber group's hands".
Sapstead said she was "incredibly concerned and disappointed" that personal data of athletes had gone into the public domain in the way it has, describing it as "incredibly distressing".
"I simply can't condone this kind of action," she added.
"I don't think it's appropriate to publicise an athlete's personal medical condition or the substance they are being subscribed."
No issues with leak - Froome
The disclosure of medical files for Britain's Froome and Wiggins by computer hackers has been condemned by British Cycling, which said it is "very proud" of its anti-doping culture.
Team Sky rider Froome, 31, said he had "no issues with the leak".
A spokesman for Wiggins said: "There's nothing new here."
The hacking group, allegedly from Russia, released the records on Wednesday, but there is no suggestion the athletes are involved in any wrongdoing.
'This leak is aimed at undermining Wada'
A British Cycling spokesperson said: "As the national governing body for the sport in Britain and a supporter of the Wada code, we condemn the publication of any individual's medical information without their permission."
Three-time Tour de France winner Froome previously said he is a "clean rider" following a hack on Team Sky's computers in July 2015.
He said on Thursday: "I've openly discussed my TUEs with the media and have no issues with the leak, which only confirms my statements.
"In nine years as a professional, I've twice required a TUE for exacerbated asthma. The last time was in 2014."
Team Sky said in a statement that applications made for TUEs have "all been managed and recorded in line with the processes put in place by the governing bodies".
Wiggins' spokesman added: "Everyone knows Brad suffers from asthma; his medical treatment is BC and UCI approved and like all TeamGB athletes he follows Wada regulations to the letter.
"The leak of these records is an attempt to undermine the credibility of Wada and that's something for them to deal with."
England women's rugby player Heather Fisher, who was also named by Fancy Bears, said she was a clean athlete who "had no issue with the information being in the public domain".
British golfer Charley Hull, who tied for seventh at the 2016 Olympics, and rower Sam Townsend, who helped GB finish fifth in the men's quadruple sculls in Rio, were also named.
British Rowing said Townsend, 30, has always worked within, and been compliant with, anti-doping rules and British Rowing take a very clear stance on this matter.
Andy Parkinson, British Rowing's chief executive, said: "Our anti-doping processes and procedures are robust and are compliant with Ukad and Wada rules. TUEs are there to support the clean athlete's right to compete."
What are therapeutic use exemptions?
A TUE allows an athlete to take a prescribed substance or undergo treatment which is prohibited.
They are frequently used because an athlete may have an illness which requires them to take certain medications without breaking anti-doping regulations.
British athletes must contact their national governing body or follow UK Anti-Doping (Ukad) guidance on whether to apply for a TUE before or after a drugs test.
There are strict criteria for one to be granted:
- The athlete would suffer significant health problems without taking the substance
- It would not be significantly performance-enhancing
- There is no reasonable therapeutic alternative to its use
- The need to use it is not due to prior use without a TUE
Ukad says it has "a number of robust controls in place to make it as difficult as possible" for athletes to misuse the system.
Why is Fancy Bears doing this?
The hacking group claims on its website that it stands for "fair play and clean sport" and that TUEs are "licenses [sic] for doping", describing Wada as "corrupt and deceitful".
Wada director general Olivier Niggli said there was "no doubt" that the hack was retaliation against Wada for its report into Russian state-sponsored cheating and appealed to the Russian government to help stop it.
Russia's track and field team were banned from the Rio Olympics over an alleged state-backed doping programme. All Russian athletes are barred from the ongoing Paralympics.
Russian authorities have denied any involvement with the hacking group.
Do they have a case?
Critics of TUEs claim they can be used by athletes as a way to bend anti-doping regulations.
A Cycling Independent Reform Commission report in 2015 claimed that abuse of TUEs is commonplace, with one rider saying 90% of them are used to boost performance.
In March 2015, former Olympic road race champion Nicole Cooke criticised both Froome for his use of TUEs and Brian Cookson, the head of the International Cycling Union, for failing to tackle the issue.
However, every athlete whose details have been leaked had the permission of the anti-doping authorities to take the substances revealed by Fancy Bears.
Froome took the steroid prednisolone to treat chest infections, while Wiggins took salbutamol to treat asthma.
Both have spoken strongly in the past about their anti-doping beliefs.