The Sheffield 33-year-old became unwell on August 6 after watching the opening ceremony of the Olympic Games, having arrived in Rio de Janeiro a few days earlier.
Webster, a former Sky and ITV sports presenter, had cycled 1,600 miles (2,575km) from Recife in the north-east of the country to Rio in 39 days with Mr Tomlinson, the widower of breast cancer fundraiser Jane Tomlinson.
She was admitted to hospital, where doctors initially thought she was dehydrated following the gruelling six-week Ride To Rio challenge.
However, her condition deteriorated after she developed a severe complication from a bacterial infection.
On Thursday, a message posted on her Twitter page read: “Thank you very much for all your messages of support, it means so much. #TeamCharlie.”
She appeared to be in good health when she arrived in the Brazilian capital on August 4 and was due to stay on in the city as part of Team GB’s Great Britons campaign alongside DJ Nick Grimshaw and model Jodie Kidd.
The marathon runner shared her frustration with her fans on social media the day after she was admitted to hospital, writing: “6 weeks on the road ends in this. Very rough day, severe dehydration & infection - 2 drips & antibiotics #Rio”.
But Webster remained positive, telling fans on August 9: “I’m getting there ... awful few days with serious infection #Rio.”
She posted a photo showing her lying in a hospital bed hooked up to a drip, but smiling, adding later: “Looking forward to being a part of @TeamGB here at #Rio2016.”
However she made no further posts and a statement on her Facebook and Twitter profiles on August 11 revealed that her condition had worsened.
It said: “Charlie has updated you all throughout her cycle and we know her silence was causing concern.
“On Saturday, 6th August, Charlie became unwell and was admitted to hospital in Rio de Janeiro.
“What was initially thought as dehydration, caused by her riding to Rio, has since been diagnosed as a severe complication caused by a bacterial infection.
“Charlie is receiving great care and we will update when we can.”
Webster, who once dated Downton Abbey actor Allen Leech, and Mike Tomlinson set off on the first leg of the Ride To Rio challenge on June 27, cycling 1,300 miles (2,092km) from London to Lisbon.
They cycled 90 miles (145km) a day with only one rest day each week before arriving at the Maracana stadium on August 4.
The Ride To Rio campaign wished Webster a speedy recovery via Twitter, describing her as a “tough lady”.
A Foreign Office spokesman said: “We are providing support to a British woman who is in hospital in Rio.”