Totally understand that you want to pursue this interpersonally before anything legal. However, no matter what your approach is, I strongly recommend that you properly understand your legal position before you approach them. Even if you don't want to "get legal" with them in your opening gambit, you really don't know how they will respond to it and you might need to fall back on your legal position. In my opinion (and based on experience of very similar situations) the worst thing is not necessarily your neighbour refusing to act - they may offer a solution that is not adequate to you.
For your introduction, go over there and say:
Hi, I'm your new neighbour, my name is Mark, my family are etc etc.
Properly introduce yourself before you get to business. Allow them to respond. Ask their name if they don't give it, remember it, and use it during the rest of the conversation - using someone's name when you are on good terms is a sign you are interested in them; but when someone is not interested in being helpful using their name can show you mean business and will follow up your words with actions. No matter what happens, using their name will send the right message.
Go on to say:
Sorry to come to you with a problem on our first meeting, but I've found some damp in my garage and it looks like it is being caused by your patio blocking the damp course.
Make sure you speak of the damp problem in the present tense (ie it "is being caused by" and not in the past like it is historic).
Now, for whatever happens next you need to be in the know legally speaking.
I am not an expert in the law, but your legal position covers at least the following:
- If any space/land/driveways between your property are "shared" your title deeds will outline what your access rights are and may well also detail what you can/can't do on it (eg build a patio on it). Property doesn't have to be leasehold to include conditions like this - freehold properties that are close together often do too, especially where paths/driveways join.
- Even if there is no "shared" land, where properties join neighbours have to grant reasonable access to one another for the purpose of repairing your own property. "Reasonable", or whatever similar word is used in law, just means you can't march onto the property and start work, you have to let them know and have some flexibility on timing.
Because you bought the property in this condition, I'm not sure if legally you can ask them to fix it and pay for it. If they are super-nice neighbours and they offer to do just that - happy days! But obviously the work has to be completed to a standard you are happy with, so be cautious about what you accept. Ask to see a detailed quotation for the work so that you know what is being done and by whom.
But, even if they don't feel like they should pay to fix it (and I don't know that they are obliged to, you'd have to ask a solicitor), you are at least legally entitled to access their property and carry out whatever work needed to fix your problem, and that doesn't include fixing up their patio to their preferred standard. So your neighbour may well be faced with a decision - do they pay to fix your damp course and make sure their patio is made good along with it, or do they allow you to rip up their patio in order to fix your damp course, leaving their patio in goodness knows what state?
Get some proper advice on this first, but I think that is the position your neighbour will be in. So if they refuse to do anything for you, just say:
Okay, well I didn't expect you to pay to repair my property, but I must get access for my workmen to access my damp course via your property. That will mean digging up whatever part of your patio is blocking it. I'll get some quotes for the work and then we can discuss a convenient time for them to carry out the work.
Then leave politely. This will give your neighbour time to think about it. They will likely reason as I have above - that it may be in their best interest to be involved in the work.