Hi Catherine, when I was diagnosed they also didn't give me anything but yes you should have Ketostix on repeat prescription (you can buy them if you have to, but everything's shut today). You can also get an Optium Xceed monitor which does both blood and ketone testing strips for fingerprick tests.
However, if you're being sick, the rules all kind of change as you can become dehydrated quickly and therefore make large amounts of ketones quickly. I was at a party once, when newly diagnosed, and there was a Type 1 girl there who told me she was given the rule that if you've been sick 3 times you need to go to A+E. I was once sick a few times (turned out to be a reaction to antibiotics). I called my GP for advice, and he told me to go straight to A+E.
So I'd suggest at the very least you call the out of hours GP service for advice but if it was me I'd be at A+E. You might end up wasting 4hrs of your time but it's better that than take a turn for the worse at home and since you have no way to test your ketones, you should let the hospital test you instead and if they say you're fine, then all's well that ends well. You can also ask them for a tub of Ketostix when you leave. Good luck!