Earlier an 18-year-old teenager was arrested in the departure area of the port of Dover in connection with the attack on Saturday morning.
Two people have been arrested as part of the probe into the attack, which injured passengers on a train at Parsons Green.
The level of "critical", the highest of the five levels used to describe the threat, means a further terrorist attack may be imminent.
Police are continuing to search a foster home in Sunbury-on-Thames, Surrey, in connection with the teenager's arrest on Saturday. He was arrested under Britain's Terrorism Act. Queen Elizabeth honored them for their efforts in 2010.
The group later released a longer statement claiming they had placed "several explosive devices" and had only detonated one of them on Friday, wounding "30 Crusaders".
"It is inevitable that so-called Islamic State, or Daesh, will reach in and try and claim responsibility".
Rudd said a key focus was to ensure that there was less material available to radicalise young people.
Images of the bomb posted on social media appear to show a bucket on fire that had been placed inside a plastic bag close to a rail vehicle door. You will see more police - including armed officers - on the streets and transport network throughout the week.
Police added that they were "keeping an open mind" about whether there may have been additional potential suspects linked to the attack.