There are 2 possibilities. Either they made a mistake or they changed it. They've made quite a few changes and Nasu said those changes are canon (an alternative universe) so it could very well be a better adaptation of their situation in London in which both Rin and Shirou spend a lot of time together.
If that's the case I'd say there are 2 reasons.
Because she's crazy possesive and selfish. I lost count of how many times she said "You're mine" in the VN, even outside her route.
This is IMO 90% of the reason. Because of Shirou's personality and circumstances. From the very beginning in the VN Shirou is doing part-time jobs to be able to pay for his daily expenses. It is stated he could stop doing them but he doesn't want to be a freeloader.
What happens in London is that he feels compelled to pay for the food, travel expenses... and he no longer has the part-time job income as he's not in Japan doing them. It would only be natural for him to look for work in London.
My guess is that in the anime Rin pretty much forced him to take the job. That way everyone wins: Rin gets to spend a lot of time with Shirou (otherwise he'd be all day working outside the magic lessons), he takes her to places to spend time together, he does things like cooking and Rin would paying for everything while they're together in London.
I'm pretty sure Shirou would have accepted even if he wasn't paid, so I wouldn't say Rin is taking advantage of him.
Edit: I've been looking for it for 15 minutes and finally found it. Just so that you know the jobs he's doing in Japan are not a breeze, this is one of them:
A simple loading job from five to eight o'clock. Even though it's only three hours, there's six hours worth of work to do. They make you run around without a minute's rest.
Moreover, Fuji-nee is constantly telling him not to arrive too late at night because of work. The only reason you don't see him working in the VN is because he gives up those part-time jobs because of the war (it makes sense...) but in reality his time is 99% school, part-time jobs and chores he does for free.