Neither Wales nor the UK are actually in a "welfare state" condition at all, but unfortunately for a lot of people in both Wales and the wider UK they - and outside observers like yourself - could be forgiven for thinking it. The UK is a rich country, but despite the ability to do so, the government we have at the moment will not spend money on services, unless its into the hands of "private providers", "private sector outsourcing", etc, where the cost to taxpayer is high and service is no better, or in many cases worse than it would be if it was provided directly by the state. And the NHS receives a very low per capita funding compared to the health services of EU neighbours.

Wales' problem is that although it has a "devolved" government, it still relies on funding being channelled to it via the UK government. The Welsh Assembly government only has very limited powers in respect of taxes. http://gov.wales/funding/fiscal-reform/welsh-taxes/?lang=en

At the moment the money that Wales receives from the EU undoubtedly comes from the UK in the first place (the UK does pay more into the EU than it gets back, the Leave campaign was correct about that, although they lied about the amounts involved). So Wales isn't being funded by other EU countries. But if the UK stops paying into the EU, and the EU stops paying Wales, the big question (I have a cynical view of what the answer will be) is whether the Westminster government will pay Wales the money that it used to get from the EU? If not, Wales will be in a big mess, especially in the rural areas that ironically voted Leave.