Can we get result using CROSS APPLY?

It is highly unlikely that you need to use a loop of any kind. My article Processing Loops in SQL Server[^] offers several worked examples with alternatives to loops.

In this case you need to identify the individual timeslots that exist on the production table e.g.

create table #timeslots (id int identity(1,1), [startDateTime] [datetime] NOT NULL,
[endDateTime] [datetime] NULL)
insert into #timeslots SELECT DISTINCT startDateTime, endDateTime FROM production

This gives each timeslot a unique identifier e.g.

1	2019-02-22 09:00:00.000	2019-02-22 10:00:00.000
2	2019-02-22 10:00:00.000	2019-02-22 11:00:00.000
3	2019-02-22 11:00:00.000	2019-02-22 12:00:00.000
4	2019-02-22 12:00:00.000	2019-02-22 13:00:00.000

All you have to do then is work out the difference in minutes between the start and end times and divide it by the number of tasks the worker was involved in in that timeslot: e.g.

select TaskId, WorkerId, T.id AS TIMESLOT
, DATEDIFF(MINUTE, T.startDateTime, T.endDateTime) / COUNT(*) OVER (PARTITION BY WorkerId, T.id)
from #production P
inner join #timeslots T ON P.startDateTime = T.startDateTime AND P.endDateTime = T.endDateTime

Results:

1	A	1	60
1	A	2	30
2	A	2	30
1	A	3	20
2	A	3	20
3	A	3	20
1	A	4	30
3	A	4	30

This will work if timeslots cross over days but won't work if the endDateTime is null. I suggest that you standardise your timeslots over all workers otherwise it is going to be difficult to interpret the results (too many timeslots representing the same period)

if anyone else wants to have a go at improving this, here is the sample data I used based on the OP's description

CREATE TABLE #production(
[id] [varchar](10) NOT NULL,
[taskId] [varchar](10) NOT NULL,
[startDateTime] [datetime] NOT NULL,
[endDateTime] [datetime] NULL,
[workerId] [varchar](5) NOT NULL
)
INSERT INTO #production (id, taskid, startDateTime, endDateTime, workerId) values
-- From 9AM to 10AM worker "A" spent 60 minutes on task 1
('1','1','22-Feb-2019 09:00:00', '22-Feb-2019 10:00:00', 'A'),
--From 10AM to 11AM "worker "A" spent 30 minutes on task 1 because worker A worked on task 1 and task 2.
('2','1','22-Feb-2019 10:00:00', '22-Feb-2019 11:00:00', 'A'),
('3','2','22-Feb-2019 10:00:00', '22-Feb-2019 11:00:00', 'A'),
--From 11AM to 12AM worker "A" spent 20 minutes on task1 because he also worked on task 2 and task 3 
('4','1','22-Feb-2019 11:00:00', '22-Feb-2019 12:00:00', 'A'),
('5','2','22-Feb-2019 11:00:00', '22-Feb-2019 12:00:00', 'A'),
('6','3','22-Feb-2019 11:00:00', '22-Feb-2019 12:00:00', 'A'),
-- From 12AM to 1PM worker "A" spent 30 minutes on task 1 because he also worked on task 3 
('7','1','22-Feb-2019 12:00:00', '22-Feb-2019 13:00:00', 'A'),
('8','3','22-Feb-2019 12:00:00', '22-Feb-2019 13:00:00', 'A')

EDIT (see OP comment below):
To get the total amount of time spent on each task you could use a CTE (or a temporary table) and just group on TaskId (note I've had to add a column name for the calculation spent)

;with cte as 
(
select TaskId, WorkerId, T.id AS TIMESLOT
, DATEDIFF(MINUTE, T.startDateTime, T.endDateTime) / COUNT(*) OVER (PARTITION BY WorkerId, T.id) as spent
from #production P
inner join #timeslots T ON P.startDateTime = T.startDateTime AND P.endDateTime = T.endDateTime
) select TaskId, SUM(spent) 
FROM cte GROUP BY TaskID

INSERT INTO #production (id, taskid, startDateTime, endDateTime, workerId) values
-- From 9AM to 10AM worker "A" worked on Task 1 and Task 2
('1','1','22-Feb-2019 09:00:00', '22-Feb-2019 10:00:00', 'A'),
('2','2','22-Feb-2019 09:00:00', '22-Feb-2019 10:00:00', 'A'),
--From 10AM to 11AM "worker "A" spent 60 minutes on task 1 
('3','1','22-Feb-2019 10:00:00', '22-Feb-2019 11:00:00', 'A'),
--From 11AM to 13:00hrs worker "B" spent 120 minutes on task 1
('4','1','22-Feb-2019 11:00:00', '22-Feb-2019 13:00:00', 'B')