Do you think a plasterer is still going to skim your walls for £150 a day when fast food workers are suddenly on £120?
Are teachers going to remain on £30k when the fast food industry is now paying just as much?
Are the solicitors working on your conveyancing going to stay on £30k now that the fast food industry paying just as much?
Why bother with any form of training when we can all just work in fast food
What do any of these professions & their salaries have to do with a potential minimum wage increase?
Teachers deserve to be paid more, nurses deserve to be paid more, fire fighters etc - McDonald's employees being paid more doesn't stop any of these things happening from being possible also. But it would lead to an increase in social mobility - that means more people spend more money to buy more things, travel more etc which leads to more money in the economy, less money being spent on benefits & government assistance, more home-owners etc.
Low-skilled, low-paid jobs & high-skilled, low-paid jobs can (& should) both earn more. This attitude that McDonald's workers don't deserve to earn more reeks (to me) of feeling threatened that people that you see as below you (i.e. the poor & uneducated), earning an amount that's similar to what you earn - and rationalising that with groups of people you don't see as below you (i.e. teachers & nurses) also being paid a low amount, therefore how dare the poor & uneducated demand more - i'm speaking generally, not aimed at you specifically.
Also yes, if more people see working in McDonalds as a viable option for work, with a low return on investment - in comparison to say going to University, or doing a trade - that makes the market more competitive. The grad market for example is ridiculous, students pay minimum 9k a year, rack up 40k worth of student loans, to get a job that pays £16-23k, often not in the field they studied at university and their wage stagnates for years - simply because grad schemes have 100's of students applying for 5 positions. That market isn't sustainable, and is part of the reason why millennials have been unable to buy houses.
edit - as an example, when I worked at Apple part-time job during Uni a lot of people in full-time were earning 28-35k (+ v good benefits), and were able to support their family, buy a house, go travelling etc, in a retail job, as opposed to going to university. More people should see this as a viable option, and if you increase minimum wage, it's possible to do that, and put less stress on the job market. That leads to higher productivity, more motivated employees, people pursuing careers because they enjoy it & have a passion for it.
All this, plus plenty more are reasons for increasing minimum wage. Wages have been stagnated for the best part of a decade under the Tories - it's created a huge gap in wealth, inflation hasn't slowed down either so the poorest in society have continued to stay poor, while the rich have continued to earn more.