All groups of Britons at the Hajj are believed to have been accounted for following Thursday's deadly crush.
The Muslim Council of Wales initially raised concerned after being unable to contact members on their phones.
But it now says all groups visiting Saudi Arabia for the religious festival have been accounted for.
British officials have been searching hospitals in the country following the stampede that killed 769 people.
Civil defence officials said more than 800 were also injured in the crush at Mina, on the outskirts of Mecca.
It happened in a morning surge of pilgrims at the intersection of streets 204 and 223 in Mina, a large valley about three miles east of the holy city.
A statement from the Saudi embassy in London said: "The majority of Thursday's victims descended onto a pathway toward (the three stone walls) Jamarat during a time that was not allocated to them."
Dilly Hussein, who has relatives at the Hajj, told Sky News: "They were so close to the incident but it managed to not affect them.
"My best friend from London actually said that he had been due to join that very group at the Mina but he realised that he was in the wrong tour group so he diverted and joined the correct group.
"Had he joined the wrong group and not realised he was in the wrong group, he could have been one of those 700 people who passed away."
Sahri Jaffri, from Manchester, tweeted: "News of deaths during the Hajj pilgrimage stampede is making me worry as my mom and her best friend are there now."
Around 20,000 Britons are making the pilgrimage, according to Rashid Mogradia from the Council of British Hajjis.
British Muslim leaders are calling for those going to the pilgrimage to be given compulsory safety training.
Saudi King Salman has ordered a safety review and investigation.
Just two weeks ago, a giant crane crashed down on Mecca's Grand Mosque, killing 111 and injuring more than 390.
In 2006, at least 346 pilgrims were killed in a stampede in the same area, and in July 1990 around 1,400 suffocated in a tunnel.
:: Relatives can call the British Embassy in Saudi Arabia on + 966 11 4819 100 or contact the FCO via Twitter @FCOtravel