when does SQL Server notice the new indexes and determines whether they actually help the query?
Immediately. Creating an index causes a recompile because of a schema change.
Here's an illustration of this. First to setup the objects in the test database:
use TestDatabase;
go
if exists (select 1 from sys.tables where object_id = object_id('dbo.TestTable1'))
drop table dbo.TestTable1;
create table dbo.TestTable1
(
Id int identity(1, 1) not null
);
go
insert into dbo.TestTable1
default values;
go 1000
if exists (select 1 from sys.procedures where object_id = object_id('dbo.GetIdFromTestTable1'))
drop proc dbo.GetIdFromTestTable1;
go
create proc dbo.GetIdFromTestTable1
@Id int
as
select Id
from dbo.TestTable1
where Id = @Id;
go
As you would expect, running this stored procedure would result in a table scan:
exec dbo.GetIdFromTestTable1 500;
go

Now let's create an index that we know this query can benefit from:
create unique index IX_TestTable1_Id
on dbo.TestTable1 (Id);
go
Re-execute the stored procedure and take a look at the post-execution plan:
exec dbo.GetIdFromTestTable1 500;
go

We can see that our table scan turned into an index seek. This shows that a new plan was generated.
Why?
We can trace this behavior with an XEvents session:
if exists (select 1 from sys.server_event_sessions where name = N'RecompileTracing')
drop event session RecompileTracing
on server;
go
create event session RecompileTracing
on server
add event sqlserver.sql_statement_recompile
(
set
collect_statement = 1,
collect_object_name = 1
)
add target package0.event_file
(
set
filename = N'RecompileTracing.xel'
);
go
alter event session RecompileTracing
on server
state = start;
go
/*
alter event session RecompileTracing
on server
state = stop;
go
drop event session RecompileTracing
on server;
go
*/
The output we get from the above illustration/repro is that we had a sql_statement_recompile
on the statement select Id from dbo.TestTable1 where Id = @Id;
(condensed with removed whitespaces/tabs/newlines) and the recompile_cause
is Schema changed.