When I moved to the UK, my accountant filed what he referred to as a "leavers" tax return which, I suppose, notifies CRA that you are no longer resident in Canada for tax purposes after a certain date. As you've moved to the UK, which has an income tax treaty with Canada, you will not have to file a further Canadian tax return after the leavers one.
I'm assuming that you have no significant assets in Canada like dividend-paying stocks, a high-interest savings account, or anything that pays you enough money to generate a T3/T4/etc slip. If you do get those, you might have to file further returns in future. You'd be best off speaking to a reputable accountant in Canada about this if you're not sure. Your run of the mill tax return shop (like H&R Block) is probably not the best place to ask as they most likely don't deal with these sorts of queries too often.
FYI, British income tax is generally covered by PAYE (pay as you earn), unless of course you've opted out or you're self-employed. As you said you're employed by someone else, it's very likely that your UK taxes are already being taken care of, so there's probably nothing at all to worry about on this end.