It sounds like you just need to find what's known as a contract manufacturer.
I actually do work for one in the UK, but as i'm here for personal interest (and learning) and not to promote the company or it's services, i'm sure you should be able to find one locally.
Contract manufactureres should in theory just do exactly as you ask and not correct your design other than sometimes physical constraints may require some modification to dimensions etc. For instance with your PCB being small it may be (will be) much better to layout multiple copies onto a panel, say into 9 or 12 boards or similar on a larger panel, which can be broken up later into the smaller individual boards.
The reason for this is, it's more cost effective to put say 10 large panels into a machine than it is to put in 100 small ones. The physical constraints being the maximim size the PCB house can produce and the maximum (or minimum) size the machines will accept when it comes to assembly etc.
Other factors are, how much hand assembly will be required?
Is it all SMT, is it all through hole, is it a mixture of both?
Single sided, double sided, PTH, solder resist, silkscreen?
Where will the components be sourced? Not all components are readily available in all countries, so equivalents may have to be sourced and the physical dimensions may be slightly different from different manufacturers.
They probably wanted the "engineer" to talk with you to try and iron out some of these simple things that many people might not consider to be an issue but design for manufacture can sometimes be a little bit different than a small prototype run.
There are dozens of bare PCB manufactures in the UK, most of them will charge you what is known as tooling costs. This is essentially producing a suitable "artwork" for their etching process and also for setting up the drilling machines etc. They will also have to perform some design rule checking on your layout to ensure that they can actually reproduce what your files are saying should be done. They will want to know if you panelise your design, what method you want used to seperate the boards from the panel. E.G. V-groove or routing beween boards.
There would be nothing to stop you going to one place to have the PCB's etched then sending them to a contract manufacturer (assembly house) to populate the boards. However it's best to talk to the assembly house first incase they have certain requirements for the boards to fit into any automatic machinery they use. IE PCB's will requre location holes and/or feducials for the alignment etc. Most assembly houses (such as where i currently work) deal directly with bare board manufacturers and know what the limitations are for each of them and they also often know who is the best quality or best price etc for the particular quantity or size of the order. I personally feel you would be better contacting them first about the assembly/population of the boards and let them deal with the actual PCB house. Of course some places do both.
As for cost, it varies between one manufacturer to the next and varies depending on all the above things i mention. They honestly will need to talk to you just to iron these things out and properly give you a quote.
I would imagine most companies won't charge you anything for a rough quote but it does depend on how much time they need to invest. It doesn't cost anything to ask them "how much will it cost for you to send an engineer out" for example but they need to asses your design to work out if your board really will need a lot of modification for production.
Where i work, we don't send out an engineer, the client usually visits us and says this is what i want, what can you do for me, so to speak. I have no idea how other companies operate obviously. Maybe there is some charge to send out the engineer but remember, often the companies are hoping to get your business so they may just chalk it up to expences hoping to get a contract out of it if they win you over. It costs nothing to ask, so ask them.