I would suggest manufacturing your product in the country you want to sell it. The reason for this is that countries like the USA and UK have strong consumer protection agencies that police Consumer goods. You cannot simply go to China, manufacter your product based on their poor standards and then sell it locally. That approach makes sense way later down the line as a cost reduction strategy.
Based on the specific product, you will have to put in the research to find "Contract Manufacturers" who can help you with your product. Be prepared, it is very expensive to get into the consumer products industries. My advice is to start very small, and focus on quality, reputation, and customer satisfaction.
Manufacturing is important, but is it the first question?
First, you need market analysis and research to quantify the opportunity.
Next, you must look at sales and distribution and understand, for instance, the questions that buyers in major retail chains will ask. How will your idea translate into value for them? How will you get your first test market?
Then, if your product (its formulation, its production process, its physical characteristics) is going to need certification on the one hand or protection on the other, you need to get this realistically costed and time-lined.
Next, how are you going to obtain samples or mock-ups of the product and appropriately designed packaging, and what about the first production runs? The UK isn't always the best place for volume manufacturing, but that isn't your immediate question. For now, you should be looking at contract manufacturers and product designers. As the UK has great quality and choice in both, start with domestic options. That will simplify communication and co-ordination, and de-risk the early period which is expensive in time and cash.
All of this presupposes that your best route is to do the whole thing yourself. For many innovators in consumer goods, the most effective route to market is either partnership with manufacturers or distribution channels, or selling the idea in to established brands. These are additional options you may choose to explore.