SQL (BigQuery) Grouping Runtime Per Day

This is an interval overlap problem. You can solve this by splitting each time period into separate days and then looking at the overlap for each day:

with t as (
      select 'Computer1' as computername, datetime '2020-06-10T21:01:28' as startdatetime, datetime '2020-06-10T21:20:19' as enddatetime union all
      select 'Computer1' as computername, datetime '2020-06-10T22:54:01' as startdatetime, datetime '2020-06-11T05:21:48' as enddatetime union all
      select 'Computer2' as computername, datetime '2020-06-08T09:11:54' as startdatetime, datetime '2020-06-10T11:36:27' as enddatetime 
     )
select dte, t.computername,
       sum(case when enddatetime >= dte and
                     startdatetime < date_add(dte, interval 1 day)
                then datetime_diff(least(date_add(dte, interval 1 day), enddatetime), 
                                   greatest(dte, startdatetime),
                                   second)
           end) as runtime_seconds
from (select t.*, 
             generate_date_array(date(t.startdatetime), date(t.enddatetime), interval 1 day) gda
      from t 
     ) t cross join
     unnest(gda) dte
group by dte, t.computername;

Below is for BigQuery Standard SQL

#standardSQL
select Date, ComputerName, 
  sum(datetime_diff(
    least(datetime (Date + 1), EndDatetime),
    greatest(datetime(Date), StartDatetime),
    second
  )) as Runtime_Seconds
from `project.dataset.table`,
unnest(generate_date_array(date(StartDatetime), date(EndDatetime))) Date
group by Date, ComputerName

if to apply to sample data in your question - as in below example

#standardSQL
with `project.dataset.table` as (
  select 'Computer1' ComputerName, datetime '2020-06-10T21:01:28' StartDatetime, datetime '2020-06-10T21:20:19' EndDatetime union all
  select 'Computer1', '2020-06-10T22:54:01', '2020-06-11T05:21:48' union all
  select 'Computer2', '2020-06-08T09:11:54', '2020-06-10T11:36:27' 
)
select Date, ComputerName, 
  sum(datetime_diff(
    least(datetime (Date + 1), EndDatetime),
    greatest(datetime(Date), StartDatetime),
    second
  )) as Runtime_Seconds
from `project.dataset.table`,
unnest(generate_date_array(date(StartDatetime), date(EndDatetime))) Date
group by Date, ComputerName

output is

Another option for BigQuery Standard SQL

Straightforward, "little silly" and almost logic-less option of just "stupidly" counting seconds in respective days - still looks like an option to me

#standardSQL
select Date, ComputerName, 
    countif(second >= timestamp(StartDatetime) and second < timestamp(EndDatetime)) as Runtime_Seconds
from `project.dataset.table`,
unnest(generate_date_array(date(StartDatetime), date(EndDatetime))) Date,
unnest(generate_timestamp_array(timestamp(Date + 1), timestamp(Date), interval -1 second)) second with offset
where offset > 0
group by Date, ComputerName

if applied to sample data from your question - output is