Collins, 66, who has difficulty walking following back surgery, suffered a severe gash near his eye after falling on the way to the toilet.
His management said he had stitches in hospital for the cut, and will be kept under observation for 24 hours.
The shows were meant to take place on 8 and 9 June and have now been rescheduled for November.
As they were postponed just hours before Collins was due to perform, some fans were already travelling to London when they heard the news.
One fan wrote on Facebook: "Sorry to hear that ... but we are sat on a train on our way to see him. #gutted .... no idea what we are going to do now ðŸ˜¢."
Another, who was at Wednesday's gig, added: " Amazing night last night!!! What a fantastic fan base you have. Get well very soon xxxxxxxx."
His management's post on Facebook added that the singer hit his head on a chair when he fell in the middle of the night, but is "recovering well". He will be kept under observation for 24 hours.
The singer, whose condition is described as "drop foot", sent "sincere apologies and thanks to fans".
The post said he has had a "fantastic week at his first shows in 10 years, cannot thank people enough for their warm reaction and is excited to return".
All tickets for 8 June will be valid for 26 November and tickets for 9 June will be valid for 27 November.
Collins will continue his tour with his next date on Sunday in Cologne, followed by Paris, before returning to London for to headline BST Hyde Park on 30 June.
Collins, who also played with prog-rock band Genesis, is one of the most successful songwriters and performers of all time.
He is one of only three recording artists, alongside Paul McCartney and Michael Jackson, who have sold more than 100 million albums worldwide both as solo artists and as members of a band.
His number one singles include A Groovy Kind of Love and Against All Odds.
Collins has won seven Grammy Awards, six Brit Awards, an Oscar and two Golden Globes.
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The company also said it had seen "no evidence" of any impact of the vote in markets outside the UK.
Its comments came as it reported an 11% rise in pre-tax profits to Â£173m for the year to 30 June.
Its profits were driven by strong trading in Asia and the rest of Europe.
In the UK, Hays said net fees were flat, with trading "more challenging" towards the end of its financial year, which coincided with the run-up to the EU referendum.
Conditions were particularly challenging in local government and healthcare markets in the UK.
Hays also said conditions were tough in banking in the City of London, and there were weakening trends in the construction and property business towards the end of the financial year.
Hays chief executive Alistair Cox said: "Following the EU referendum, there is increased uncertainty in the UK market, but we have seen no evidence of any impact elsewhere.
"It is too early to tell what the longer term impact may be and as ever, we will monitor activity levels closely."
Kean Marden, an analyst at Jefferies, said: "In construction and property, London and larger corporates have been most impacted. To date, Hays has seen no evidence of contagion into Europe."
"We are mindful that July and August are seasonally quiet months for the industry and September (which can be one of the largest revenue contributors of the year) will provide more meaningful insight."
The UK comprises about a third of Hays' business.  It employs about 9,200 people across 33 countries.
In the Asia Pacific region net fees grew by 4%, with Australia up 5%, boosted by strong public sector growth.
In continental Europe and the rest of the world, net fees increased by 15%.
Pop star Phil Collins has cancelled two shows at the Royal Albert Hall after he slipped over in his hotel room.
[NEXT_CONCEPT]
Recruitment giant Hays has said the UK job market weakened "significantly" around the time of the EU referendum, but it is too early to judge the long-term impact of the vote.