Yes, you might have urodynamics. This tests for stress incontinence. I've had it done. It is not the most pleasant thing, but it is survivable and at the very least it would rule stress incontinence out for you, if it is not that. An overactive bladder can result in a thickened bladder wall, and this should show up on ultrasound.
If nothing physically traumatic happened to you in the run up to that 6 months prior to your pregnancy, it seems strange for you to suddenly develop stress incontinence. No sudden weight gains? No bodybuilding and lifting massively heavy weights for a protracted period of time? Yes, after the birth of your DC, your pelvic floor might very well be weaker, but you had problems long before then, so it seems like it just can't be the whole issue.
Has your wee been tested for bacteria? Long standing infection can also cause problems.
As I say, I've had both types. With stress incontinence, there was that sudden urgency which I couldn't control. It felt like it does when you have cystitis and need to pee all the time. With the urge incontinence, I was leaking with far fewer feelings of needing to go, IYSWIM. With the stress incontinence it felt like IF ONLY I could clench or something, I could stop the flow. With urge incontinence it felt like there was nothing I could do to stop the flow. It was just going to happen. Does that make any sense? What does it feel like for you?
I know how distressing it is. I never thought about any of this until it happened to me. As others have said, it CAN be sorted, and comparatively easily and with good results. The first step was to go to your GP and for them to take it seriously. Well done on taking that first step and it's great you are being referred.
BTW when you have your appointment, keep emphasising that the problem started 6 months prior to conception. They need to divorce this problem from the fact that you have recently given birth! Can you leave your baby with someone when you go to hospital (only half joking....!)?