Starting from September the number of Covid-19 positive cases has been constantly growing in London, as we can see in the graphic below, published by the Greater London Authority
There is a growing number of positive cases among youngsters too. However, the number of patients requiring hospitalization is extremley low, when compared to the first wave in Spring.
Because of this second wave there are more restrictions in London, like the opening times of bars and restaurants, the number of people allowed in closed spaces, etc.
For many travellers there is also the inconvenience of having to quarantine for 14 days on arrival, and this makes travelling not advisable at present.
The list of countries that do not have to quarantine is constantly updated, and can be found on the government website (https://www.gov.uk/guidance/coronavirus-covid-19-countries-and-territories-exempt-from-advice-against-all-but-essential-international-travel).
The list below has been updated on the 15 October
Faroe Islands and Greenland
Greece (not including Mykonos)
Madeira and the Azores (not including mainland Portugal)
Antigua & Barbuda
British Virgin Islands
South Georgia & the South Sandwich Islands
St Kitts and Nevis
St Pierre and Miquelon
St Vincent and The Grenadines
British Indian Ocean Territory
Wallis and Futuna
St Helena, Ascension and Tristan da Cunha