Patients have contacted the BBC to say they have had problems getting their Tamiflu supply.
Sainsbury's have said they will be opting out of distributing Tamiflu over fears it might encourage swine flu victims into its supermarkets.
Tesco and Asda have both signed up some of their pharmacies as collection points for people needing the drug.
The Conservatives have criticised the National Flu Service, claiming the system is "too little, too late".
Tory health spokesman Mark Simmonds said the web and telephone service should have been set up earlier when a global pandemic was declared.
The National Flu Service offers sufferers access to drugs on the phone and via the internet without the need to see a GP.
The government said demand was "unprecedented", with 2,600 hits a second. The site crashed shortly after its launch for about two and half hours.
Chief Medical Officer Sir Liam Donaldson said they had increased capacity but expected demand to drop.
Sir Liam Donaldson: "There will be an element of the worried well"
"Our initial analysis suggests many people visited the site out of curiosity, they started to check the checklist and then abandoned it.
The Tories said there was "still far too much confusion about access to diagnosis and treatment for swine flu".
Mr Simmonds told BBC Radio 4's Today programme: "We should have had the flu line and the website up earlier. We should have gone live when the WHO [World Health Organisation] said we had pandemic levels.
"Indeed, the fact that the website crashed within two to five minutes of it being up and running, I think, is a function of the fact that it wasn't up earlier.
THE PANDEMIC FLU SERVICE
People in England can phone 0800 1 513 100 or log on to www.direct.gov.uk/pandemicflu
People in Wales, Scotland and Northern Ireland should contact NHS Direct or their GP
The network of call centres is being staffed by 1,500 people to start with working from 8am to midnight
They are using a checklist to diagnose swine flu and can issue a voucher number if drugs are required
The public is also able to use the website to fill in the checklist and get access to the drugs that way
Sufferers are still able to go direct to GPs and pregnant women, parents of children under one and those with underlying health conditions should all contact a doctor if swine flu is suspected
"Therefore, people couldn't access the information, they couldn't discover themselves whether they had the appropriate symptoms that would give them the opportunity to get Tamiflu.
"I think it is a direct result of that."
The latest UK death toll from swine flu stands at 26.
The DoH said two-thirds of the deaths were among people with severe health conditions, such as cancer.
Some 840 patients are in hospital - 63 of which are in critical care - compared to 652 last week.
Dr Ari Ercole, University of Cambridge, an author on the study on critical care bed capacity said provision in paediatric intensive care units would be "quickly exhausted."
He added that there could be a 20% excess demand for ventilators in some areas.
A spokesman for the Department of Health said all NHS organisations had pandemic plans in place.
"As part of our preparations, guidance has been issued which contains information for primary and secondary care services in the UK on managing surge capacity and the prioritisation of services and patients during an influenza widespread outbreak."
To contact the National Flu Service, people in England can call 0800 1 513 100 or log on to www.direct.gov.uk/pandemicflu
The flu service is not covering the rest of the UK as Wales, Northern Ireland and Scotland have all experienced much less demand.
People with underlying health problems, pregnant women and parents concerned for children aged under one are still being advised to contact a GP.
A checklist is being used by call centre staff and the website to decide whether people have swine flu.
If the virus is suspected, people will be given a voucher number to give to a flu friend to take to a collection point to pick up anti-flu drugs.
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