Currently, mortgages are 4 to 4.5 times your salary, so people don't overstretch themselves; that's not a £350k house for you. They will also look at where the deposit came from; there's a big difference between saving 20k and 30k and do your parents need repaying?
It's not 20 years of a consistent £2k a month. A typical mortgage lasts 25 years, but within that you can have deal options, eg you pay a variable rate for 5 years, then a fixed rate for 5 years. You have to renew your mortgage and move to another deal at the end of each 5 years (5 as an example, could be shorter, could be longer) and it doesn't necessarily have to be with the same provider. There are options to overpay and pay off early, but beware of fees; if you can overpay, it's better putting the money into something earning extra interest and paying extra when there's no fees due, which is normally after a few years.
Have you factored in the fees for buying a house? Some you could add to your mortgage rather than pay seperately. And all the costs? For the latter there's the upfront expense of furniture, soft furnishings, white goods, electrical appliances. Then bills: council tax, insurance, utilities, broadband, phone, TV licence. Then there's personal expenses, including travel, depending on where you buy, food, clothes, laundry and cleaning, gardening, socialising, credit cards, hobbies, subscription services, holidays.
Me: 10% deposit, 25 year mortgage, paid it off in 12 years. Used Fixed Rate Bonds as short-term investments for any spare cash and overpaid the mortgage when it was free to do so. Had it fairly well worked out on a spreadsheet as my provider advertised its deal options well in advance and I could adjust my figures with interest rate changes and overpayments. My calculations were within about £400 of the final lump sum that I paid to clear the mortgage, approx £8000.