In the interim make sure that you keep notes and photographs of any problems and damage caused.
It's potential grounds for legal action but always better to come to an amicable solution especially if it's with your neighbours.
Initially I'd say give them a reasonable time frame to have it rectified or you will have to consider taking further action as it is causing a great deal of inconvenience, damage to your property and impact in value.
I'd make it for before you go away on your holiday stating that you can't possibly go away and risk further damage to your property. It gives them over a month which, depending on the work required, sounds reasonable. Make sure there is a gap before you go to make sure any work done has actually worked.
You could also note that, if you are forced to cancel your holiday then you will also be looking for recompense for that through the courts should you have to take the matter further.
Keep copies of everything.
Is there any way you can protect your conservatory in the meantime? If you post in Home and Garden then people there may have some possible solutions for you.
Just make sure you stress that any measures are temporary so they don't use it as an excuse to waste any more time and I'd suggest the neighbours help fund the cost.
Is there someone you can get to check on your property while you are away just in case?
On a legal point, would be interested to see if your deeds include a right of natural drainage etc...
If there is anything that planning would have had to have got involved in then you could try speaking to the local council about it. Even it's it's not in their remit they may be able to provide some advice and assistance.