From Monday 18th January people entering England from abroad (including UK citizens) will need to provide evidence of having a negative COVID-19 test which was taken in the 3 days prior to starting their journey. There are a few exemptions for this rule including children aged 11 who do not need to take a test. For full details, including the types of test accepted, please visit the UK Government website here.
Visitors travelling from countries listed on the UK Government's Travel Corridor List will not need to self-isolate when they enter England (but will still need to have taken the negative COVID-19 test).
For visitors arriving from countries not listed on the Travel Corridor List, in addition to taking a negative COVID-19 test before travelling, they will need to self-isolate for 10 days upon arrival. This 10 days can be shortened by taking a private COVID-19 test 5 full days after leaving the country not on the Travel Corridor List. If this test proves negative they will no longer need to self-isolate. The private test should be booked before travelling. Full details of this scheme, known as the Test to Release for International Travel Scheme, can be found here.
I hope the above information helps. As we move through the pandemic I suspect these requirements will change so please do check the UK Government's website for the latest advice before travelling.
UPDATE (16th January):
The Government has announced that from Monday 18th January the Travel Corridor scheme will be suspended. This means that all arrivals into England will have to self-isolate for 10 days, no matter where they are travelling from. As described above, this 10 days can be reduced by taking a private test after 5 days. The Travel Corridor scheme is expected to be re-introduced after England's national lockdown finishes - which is expected to be mid-end February. From Monday all arrivals will still need evidence of having a negative COVID-19 test prior to travel, as mentioned above.