just get your new internet deal with someone else to start a day or two before your BT one officially ends, that way you won't have any down time.
£50 a month seems pretty high I went with virgin and they threw all kinds of offers at me so i ended up with fiber plus they matched my giffgaff phone contract to switch to O2 for a combined £28 a month (for the first 18 months then it goes up, at which point I will switch again) with £100 pound credit.
Don't just stick with the same company for convenience, thats what they are banking on.
