It depends a lot on your individual course/university. Some departments/courses, as noted above, provide a fixed amount of printing credit every year. Others do not. Even within individual universities this varies - at Exeter, UG Physicists had I think £10-20 of credit each year (I think they got more in 3rd/4th years) while the engineers had none. At Southampton in Maths we had I think £5 in first year.
You can get reasonably cheap printers these days, although the return on investment for them may be less good than you think due to poorer quality and similar issues. I would note that for BMS, you will probably have fairly regular labs, for which you will be expected to write lab reports. At the very least, you will probably need to print a cover/submission sheet for each one, if not typesetting the whole report. As such, it may be a good idea to get one since you may do more printing than other courses/you expect.
It may be worth considering a printer with a built in scanner or photocopier as well, as you might be expected to record results in the lab itself but include these rough results in your report somehow - it may be acceptable to include a scanned version, which if so would probably better than stapling a possibly by the point of handing in quite battered bit of graph paper...however your mileage may vary on this. On the photocopying side, you may want to photocopy sections of notes to annotate or create a copy of your/a friends notes if one of you misses a lecture due to illness or similar.
In all cases, I would suggest waiting until you get to the uni and have your induction to find out what facilities you have available to you. You can probably get one Amazon Prime-d if need be in the first/second week, along with paper and ink, ensuring you don't get something that is more expensive and has more features than needed or less so on either front.