It’s better to have one, most companies ask you to open one in order to be paid, nowadays almost employer pay wages directly into bank account.

Unlike most countries, in the UK you can, according to your bank, get a free bank account with a debit card.

• Some banks  do not ask for the standard verification as proof of address. Call in to the bank and see what you need to bring in with you before you apply.
• If your employer has provided you with a contract of employment or payslip with your current address on it, you may beable to use that. 
• If you don't drive, apply for a provisional driving license at the post office (check that you can qualify first). There is a charge for this and the DVLA takes around 2 weeks to send this through to your address. You can then use this as a valid proof of your address.
• Speak to the person you're living with. Ask to take on one of the bills in your name. 
• Apply for a Prepaid Card. These cards are great for students and those who have little or no credit history.
  Once the card is sent to your home, use the letter that comes with it as proof of address.

It depends on your own skills and English level, obviously if you already are fluent in English the chances will be a lot more. (see the question about the knowledge of English to discorve more).
Even the kind of job that you're looking for will influence the waiting period.

Check the Curriculum Vitae page.

A cover letter is a document sent with your resume to provide additional information on your skills and experience.
You should write one if your are applying via online form; Iit provides detailed information on why you are qualified for the job you are applying for and explains the reasons why your are interested in that job.
Have a good cover letter helps you to obtain an interview with employers, even if an employer doesn't request a cover letter, it's helpful to send one.

Tips:

• Targ the cover letter according to the job
• Do not copy your CV, extend it
• Address it to a specific person, if it is possible
• Keep it simple, don't overdo.

Speaking English immediately opens up opportunities to be employed quicker than a non-english speaker and will open your job prospects and increase your standard of living, working knowledge of the English language can create many opportunities in international markets and regions.

If you are a complete beginner than we suggest to have one or two weeks of English school before start searching for a job.

Check your level with these two level tests:

Grammar & vocabulary level test

Listening level test

At the end of each test your level will be assessed at a CEF level (A2 to C2).

The UK school system is very different form the European one, a high school diploma as the "Baccalauréat" is an "A-Level" equivalent.
for further information check the website: www.gov.uk/what-different-qualification-levels-mean

A good connection to the internet is provided by most London accommodation, but if you're out there are many cafés providing internet access for an average price of £1 per hour.

You can even access Wi-Fi almost anywhere in London, in coffee shops, bars, restaurants, libraries and some tube stations. Simply ask the passcode to a member of the staff.