The data from Public Health England is based on the number of people testing positive in either a lab-reported or rapid lateral flow test for the seven days to May 22.
Of the 315 local areas in England, 137 (43 per cent) have seen a rise in rates, 162 (51 per cent) have seen a fall and 16 are unchanged.
It comes as a total of 3,180 new UK cases of Covid-19 were reported on Wednesday, the highest daily number since April 12. However, it is too soon to say whether this is part of a consistent upwards trend.
The Covid rate in Hounslow, that has been highlighted as a hotspot for the Indian variant, is said to be the highest in London, with 53.8 per 100,000 people. There were 146 new cases but the rate and number of cases has fallen slightly compared to the previous week.
Neighbouring Hillingdon saw a 47 per cent increase to record 164 additional positive tests between May 15 and May 22, the Public Health England data suggests. It recorded 53.4 cases per 100,000 people, the second highest rate in London.
The number of cases in Lambeth more than doubled to 147. Its Covid rate also more than doubled from 21.8 cases per 100,000 people to 45.1.
Ealing saw weekly positive coronavirus tests fall to 153.
Kingston-upon-Thames health officials recorded a 22 per cent increase to 71 cases.
In Newham, they rose a third to 64 and in Hackney and City of London went up by more than half to 41.
Croydon saw a third reduction to 149 positive tests. Waltham Forest saw a 24 per cent increase to 41.
Brent, Richmond upon Thames, Hammersmith and Fulham, Greenwich, Kensington and Chelsea, Islington, Sutton, Crawley and Watford saw cases pretty much unchanged.
Some London boroughs saw a fall in cases. There was a 39 per cent reduction in Lewisham to 37 cases. Officials in Tower Hamlets recorded 38 positive tests, a drop of one fifth. Barking and Dagenham had just 20 cases, a 35 per cent decrease.
Data for the most recent four days May 23-26 has been excluded as it is incomplete and does not reflect the true number of cases.
Bolton in Greater Manchester continues to have the highest rate, with 1,286 new cases in the seven days to May 22 – the equivalent of 447.2 cases per 100,000 people.
This is up from 321.7 in the seven days to May 15.
Blackburn with Darwen in Lancashire has the second highest rate, up from 145.0 to 313.3, with 469 new cases.
Bedford has the third highest, up from 127.0 to 202.0, with 350 new cases.
The number of UK-wide cases reported each day has been broadly flat for the past month, sometimes dipping below 2,000 (most recently on May 17) but mostly staying between 2,000 and 3,000.
The seven-day rolling average has crept upwards slightly in recent days, but more data is needed to see whether this is the start of a continuous nationwide increase and not simply a short-term response to a spike in cases in a few hotspots.
Separate data for cases by local area suggests that only a small number of places in the UK are currently recording a rise in cases that is greater than the usual day-to-day fluctuation in numbers.