Issue with using datetime with dates on either side of clocks going forward/backward

From what I observed (in Germany), the situation is as follows.
DatetimeRangePicker assumes that the programmatic input time is UTC and converts it to local time before showing the times in the GUI.
In Germany, in the summer time (2019-09-20 in your example) a programmatic input of 00:00:00 will be displayed 02:00:00 by DatetimeRangePicker.
Since in 2019 the switch from daylight saving time to winter time was done on October 27th in the morning, for October 28th, 00:00:00 will be displayed 01:00:00 by DatetimeRangePicker.

May I ask in which time zone you are? From the video you posted I would guess the same time zone as UK.

Interestingly, in the converse way, no conversion is done. So if I choose a start time of 02:01:00 in the GUI of the DatetimeRangePicker, it returns 02:01:00.

This is a somewhat strange behavior of DatetimeRangePicker. I don’t know if it can be tuned by any arg (maybe from the Javascript side).

Also, be aware that as far as I know, Bokeh doesn’t support time zones, any information about time zone and daylight saving time will be ignored:

So localizing with pandas up front will not directly help.

Though, you can manage the situation by applying following trick:
Before giving any time value to DatetimeRangePicker, transform it to a fake UTC for instance with:
pd.Timestamp(time, unit="ms", tz="Europe/Berlin").tz_convert("utc")
using your time zone as tz-arg.

Rewriting your code could lead to:

from bokeh.plotting import curdoc
from bokeh.models import DatetimeRangePicker, Select
from bokeh.layouts import row
import pandas as pd

# Helper fonction:
def fake_utc(time):
    return pd.Timestamp(time, unit="ms", tz="Europe/Berlin").tz_convert("utc") # Use your time zone arg

# Helper boolean to identify programmatic changes to datetime_range_picker:
prog_change = False

date_options = ["2019-09-20", "2019-10-26", "2019-10-28"]

date_select = Select(
    title="Select date",
    value=date_options[0],
    options=date_options
)
def date_select_cb(attr, old, new):
    # global datetime_range_picker # global declaration not necessary because only attributes are changed
    global prog_change
    print(f"date_select_cb(attr, old, new) called, new: {new}")
    prog_change = True
    # datetime_range_picker.visible = True
    datetime_range_picker.min_date = fake_utc(new + "T00:00:00")
    datetime_range_picker.max_date = fake_utc(new + "T23:59:59")
    datetime_range_picker.value = (fake_utc(new + "T00:00:00"), fake_utc(new + "T01:00:00"))

date_select.on_change("value", date_select_cb)

datetime_range_picker = DatetimeRangePicker(
    title="Select time range",
    min_date = fake_utc(date_select.value + "T00:00:00"),
    max_date = fake_utc(date_select.value + "T23:59:59"),
    value = (fake_utc(date_select.value + "T00:00:00"), fake_utc(date_select.value + "T01:00:00")),
    width=400,
)

def datetime_range_picker_cb(attr, old, new):
    global prog_change
    if prog_change:
        print(f"\ndatetime_range_picker_cb(attr, old, new) called due to a programmatic change of datetime_range_picker.value")
        print(f"new (as fake utc): {new} = ({pd.Timestamp(new[0], unit="ms")}, {pd.Timestamp(new[1], unit="ms")})")
        prog_change = False
    else:
        print(f"\ndatetime_range_picker_cb(attr, old, new) called due to a manual change of datetime_range_picker.value")
        print(f"new: {new} = ({pd.Timestamp(new[0], unit="ms")}, {pd.Timestamp(new[1], unit="ms")})")
        # This is what you probably want to have.

datetime_range_picker.on_change("value", datetime_range_picker_cb)

layout = row(date_select, datetime_range_picker)

curdoc().add_root(layout)

Then in my case, DatetimeRangePicker shows the time I introduced up front, without offset in summer or winter time.
But I would expect that this solution could still lead to an offset for a few hours around the time changing hour in October and March.