Along the lines of raising a minimum wage and the products around it raising in price, this change changes the cost of business. When you can pay your drivers less you make more profit as a company, but when they demand more your only options are really to either radically change the way you work your party scale, or increase the price of doing business. When split between lower profit or "happier employees/customers" a business will almost always choose the route that maximizes profit while potentially ignoring both.
Yes, it's very possible that Uber fees will increase now that drivers in the UK have won the right to earn minimum wage and get holiday pay. But the increase will likely be small. But even if the increases are significant, it's important that workers make a living wage. It's wrong to expect people to work as virtual slaves.
This is always the case. The cost of better pay and better benefits for employees will always be passed along to the consumer. This is why we can't jump to a $15 minimum wage across the board. Either the cost will be passed along to consumers or employers will make cuts to the work force. In the case of Uber, you can't really make cuts to the workforce since they are independent contractors, so the cost will be passed on to the consumer meaning higher fees. Then demand will decrease because of higher fees so fees will increase more. It's a never-ending cycle.
Uber may have to take less from its drivers, but if there's too much of a fee increase, it will driver customers away. No reason Uber drivers can't have holiday pay or make minimum wage. I'm sure there's a way to balance the drivers' need with the company's desire for profit.