It could be something as simple as stress incontinence. Say, for example, you cough or turn over in your sleep and you leak.
Pregnancy and all the hormones can make it harder for your pelvic floor to function correctly.
It's also possible that you have a UTI which would make you more prone to leaking. Get this checked out.
One thing is for sure, your kidneys and bladder go into overdrive a bit in the early stages of pregnancy, which is why women go for a wee more often at that stage. Try to avoid drinking anything that could further aggravate your bladder like caffeine, citrus drinks and blackcurrant.
I would go to the doctor or midwife, tell them exactly what you said here. They can check your wee and discuss all the reasons why this has started happening. Don't be ashamed of it, the doctors/midwives hear about this kind of thing all the time.
In some areas you can make a self referral to your local continence nurse, might be worth googling to see who yours is or ask your GP.
Either way, get going on some pelvic floor exercises if your haven't already. Make sure you drink plenty ie 8 glasses a day (don't avoid drinking to avoid the problem, this doesn't work and you need to keep yourself well hydrated in pregnancy) You could also try to ensure you've drank all you need a good three hours clear before bedtime.
I'm 20 weeks and have gone through various strange bladder symptoms. All of them have come and gone but one of them which has persisted from about week 5 is the need to get up for a wee 1-2 times a night which I absolutely NEVER needed to do prior to pregnancy.
So there's a chance this bed wetting is down to your hormone fluctuations and will eventually settle and go away too.