#!/usr/bin/env python
# -*- coding: utf-8 -*-

import unicodedata
text = u'Cześć'
print unicodedata.normalize('NFD', text).encode('ascii', 'ignore')

For converting to ASCII you might want to try ASCII, Dammit or this recipe, which boils down to:

>>> title = u"Klüft skräms inför på fédéral électoral große"
>>> import unicodedata
>>> unicodedata.normalize('NFKD', title).encode('ascii','ignore')
'Kluft skrams infor pa federal electoral groe'

The data for the .ics file should not be decoded, but passed directly to .from_ical. Use res.content instead. Then Calendar generates the data decoded correctly as UTF-8 (probably part of the .ICS spec) and print can print Unicode strings correctly. For the JSON, write with utf8 encoding and ensure_ascii=False as @JosefZ recommended to see it correctly as well:

import requests
import json
from icalendar import Calendar

url = 'http://www.formula1.com/calendar/Formula_1_Official_Calendar.ics'
res = requests.get(url)
calendar = Calendar.from_ical(res.content)
events = [
    {
        'id': event['UID'].split('@')[-1].strip(),
        'startTime': event['DTSTART'].dt.strftime('%Y-%m-%dT%H:%M:%S.%f')[:-3],
        'summary': event['SUMMARY']
    } for event in calendar.walk('VEVENT') if str(event['UID']).split('@')[0].startswith('Race')]

for event in events:
    print(event['summary'])

with open('events.json', 'w', encoding='utf8') as f:
    json.dump(events, f, ensure_ascii=False, indent=2)

print Output:

FORMULA 1 GULF AIR BAHRAIN GRAND PRIX 2021 - Race
FORMULA 1 PIRELLI GRAN PREMIO DEL MADE IN ITALY E DELL'EMILIA ROMAGNA 2021 - Race
FORMULA 1 HEINEKEN GRANDE PRÉMIO DE PORTUGAL 2021 - Race
FORMULA 1 ARAMCO GRAN PREMIO DE ESPAÑA 2021 - Race
FORMULA 1 GRAND PRIX DE MONACO 2021 - Race
FORMULA 1 AZERBAIJAN GRAND PRIX 2021 - Race
FORMULA 1 HEINEKEN GRAND PRIX DU CANADA 2021 - Race
FORMULA 1 EMIRATES GRAND PRIX DE FRANCE 2021 - Race
FORMULA 1 MYWORLD GROSSER PREIS VON ÖSTERREICH 2021 - Race
FORMULA 1 PIRELLI BRITISH GRAND PRIX 2021 - Race
FORMULA 1 MAGYAR NAGYDÍJ 2021 - Race
FORMULA 1 ROLEX BELGIAN GRAND PRIX 2021 - Race
FORMULA 1 HEINEKEN DUTCH GRAND PRIX 2021 - Race
FORMULA 1 HEINEKEN GRAN PREMIO D’ITALIA 2021 - Race
FORMULA 1 VTB RUSSIAN GRAND PRIX 2021 - Race
FORMULA 1 SINGAPORE AIRLINES SINGAPORE GRAND PRIX 2021 - Race
FORMULA 1 JAPANESE GRAND PRIX 2021 - Race
FORMULA 1 ARAMCO UNITED STATES GRAND PRIX 2021 - Race
FORMULA 1 GRAN PREMIO DE LA CIUDAD DE MÉXICO 2021 - Race
FORMULA 1 HEINEKEN GRANDE PRÊMIO DE SÃO PAULO 2021 - Race
FORMULA 1 ROLEX AUSTRALIAN GRAND PRIX 2021 - Race
FORMULA 1 SAUDI ARABIAN GRAND PRIX 2021 - Race
FORMULA 1 ETIHAD AIRWAYS ABU DHABI GRAND PRIX 2021 - Race

events.json:

[
  {
    "id": "1064",
    "startTime": "2021-03-28T16:00:00.000",
    "summary": "FORMULA 1 GULF AIR BAHRAIN GRAND PRIX 2021 - Race"
  },
  {
    "id": "1065",
    "startTime": "2021-04-18T14:00:00.000",
    "summary": "FORMULA 1 PIRELLI GRAN PREMIO DEL MADE IN ITALY E DELL'EMILIA ROMAGNA 2021 - Race"
  },
  {
    "id": "1066",
    "startTime": "2021-05-02T15:00:00.000",
    "summary": "FORMULA 1 HEINEKEN GRANDE PRÉMIO DE PORTUGAL 2021 - Race"
  },
  {
    "id": "1086",
    "startTime": "2021-05-09T14:00:00.000",
    "summary": "FORMULA 1 ARAMCO GRAN PREMIO DE ESPAÑA 2021 - Race"
  },
  {
    "id": "1067",
    "startTime": "2021-05-23T14:00:00.000",
    "summary": "FORMULA 1 GRAND PRIX DE MONACO 2021 - Race"
  },
  {
    "id": "1068",
    "startTime": "2021-06-06T13:00:00.000",
    "summary": "FORMULA 1 AZERBAIJAN GRAND PRIX 2021 - Race"
  },
  {
    "id": "1069",
    "startTime": "2021-06-13T19:00:00.000",
    "summary": "FORMULA 1 HEINEKEN GRAND PRIX DU CANADA 2021 - Race"
  },
  {
    "id": "1070",
    "startTime": "2021-06-27T14:00:00.000",
    "summary": "FORMULA 1 EMIRATES GRAND PRIX DE FRANCE 2021 - Race"
  },
  {
    "id": "1071",
    "startTime": "2021-07-04T14:00:00.000",
    "summary": "FORMULA 1 MYWORLD GROSSER PREIS VON ÖSTERREICH 2021 - Race"
  },
  {
    "id": "1072",
    "startTime": "2021-07-18T15:00:00.000",
    "summary": "FORMULA 1 PIRELLI BRITISH GRAND PRIX 2021 - Race"
  },
  {
    "id": "1073",
    "startTime": "2021-08-01T14:00:00.000",
    "summary": "FORMULA 1 MAGYAR NAGYDÍJ 2021 - Race"
  },
  {
    "id": "1074",
    "startTime": "2021-08-29T14:00:00.000",
    "summary": "FORMULA 1 ROLEX BELGIAN GRAND PRIX 2021 - Race"
  },
  {
    "id": "1075",
    "startTime": "2021-09-05T14:00:00.000",
    "summary": "FORMULA 1 HEINEKEN DUTCH GRAND PRIX 2021 - Race"
  },
  {
    "id": "1076",
    "startTime": "2021-09-12T14:00:00.000",
    "summary": "FORMULA 1 HEINEKEN GRAN PREMIO D’ITALIA 2021 - Race"
  },
  {
    "id": "1077",
    "startTime": "2021-09-26T13:00:00.000",
    "summary": "FORMULA 1 VTB RUSSIAN GRAND PRIX 2021 - Race"
  },
  {
    "id": "1078",
    "startTime": "2021-10-03T13:00:00.000",
    "summary": "FORMULA 1 SINGAPORE AIRLINES SINGAPORE GRAND PRIX 2021 - Race"
  },
  {
    "id": "1079",
    "startTime": "2021-10-10T06:00:00.000",
    "summary": "FORMULA 1 JAPANESE GRAND PRIX 2021 - Race"
  },
  {
    "id": "1080",
    "startTime": "2021-10-24T20:00:00.000",
    "summary": "FORMULA 1 ARAMCO UNITED STATES GRAND PRIX 2021 - Race"
  },
  {
    "id": "1081",
    "startTime": "2021-10-31T19:00:00.000",
    "summary": "FORMULA 1 GRAN PREMIO DE LA CIUDAD DE MÉXICO 2021 - Race"
  },
  {
    "id": "1082",
    "startTime": "2021-11-07T17:00:00.000",
    "summary": "FORMULA 1 HEINEKEN GRANDE PRÊMIO DE SÃO PAULO 2021 - Race"
  },
  {
    "id": "1083",
    "startTime": "2021-11-21T06:00:00.000",
    "summary": "FORMULA 1 ROLEX AUSTRALIAN GRAND PRIX 2021 - Race"
  },
  {
    "id": "1085",
    "startTime": "2021-12-05T16:00:00.000",
    "summary": "FORMULA 1 SAUDI ARABIAN GRAND PRIX 2021 - Race"
  },
  {
    "id": "1084",
    "startTime": "2021-12-12T13:00:00.000",
    "summary": "FORMULA 1 ETIHAD AIRWAYS ABU DHABI GRAND PRIX 2021 - Race"
  }
]

You need to escape the backslash so that \u isn't treated as the beginning of a Python escape sequence.

"Czesć".replace("ć", "\\u0107")

If, s=url['title'] makes s equal to this:

In [48]: s=u'Oscar Winners Best Pictures Box Set \xc2\xa36.49'

Then the problem is

1. in the code that defines url,
2. or else the content from the web is mal-formed.

If Case 1, we'd need to see the code that defines url.

If Case 2, a quick-and-dirty workaround would be to encode the unicode object s with the raw-unicode-escape codec:

In [49]: print(s)
Oscar Winners Best Pictures Box Set £6.49

In [50]: print(s.encode('raw-unicode-escape'))
Oscar Winners Best Pictures Box Set £6.49

See also this SO question.

Regarding titles like s=u'Star Trek XI £3.99': Again, it would be nice fix the problem before it gets to this stage -- perhaps by looking at how url is defined. But assuming the content from the web is mal-formed, a workaround would be:

In [86]: import re

In [87]: print(re.sub(r'&#x([a-fA-F\d]+);',lambda m: unichr(int(m.group(1),base=16)),s))
Star Trek XI £3.99

A little bit of explanation:

Note that

In [51]: x=u'£'
In [53]: x.encode('utf-8')
Out[53]: '\xc2\xa3'

So the unicode object u'£', encoded with the utf-8 codec, becomes the string object '\xc2\xa3'.

Somehow, url['title'] is getting defined to be the unicode object u'\xc2\xa3'. (The u makes a big difference!)

Thus we have u'\xc2\xa3' when we desire '\xc2\xa3'. Encoding the unicode object u'\xc2\xa3' with the raw-unicode-escape codec transforms it to '\xc2\xa3'.

1. Decode the string to Unicode. Assuming it's UTF-8-encoded:

   str.decode("utf-8")
   

2. Call the replace method and be sure to pass it a Unicode string as its first argument:

   str.decode("utf-8").replace(u"\u2022", "*")
   

3. Encode back to UTF-8, if needed:

   str.decode("utf-8").replace(u"\u2022", "*").encode("utf-8")
   

(Fortunately, Python 3 puts a stop to this mess. Step 3 should really only be performed just prior to I/O. Also, mind you that calling a string str shadows the built-in type str.)