Hi. Can anyone tell me if I can go stright on to a PhD with a first degree (without doing a Masters)?. I have a 1st in Engineering.

yes you can

Of course you can. Most students apply for a PhD with a 2.1 (hence, you have an advantage).

For UK students a first or a 2i is usually enough to go on straight to a PhD in science or engineering.

Most universities don't really give a rot about your overall result, what matters is lab experience you and what scores you got in relevant modules/the relevance of your BSc course. If you get a 1st with a written dissertation and scored badly in relevant modules, it's not going to look as good as a 2:1 student with an experimental project and relevant modules. But in short, yes, you can go straight onto a PhD with a 1st class if your BSc had relevant modules