Nobody knows any of this, especially the people who engineered the 'out' vote. They have no idea how the arrangement with Europe will pan out & will argue like fuck between themselves about it, whilst the europeans argue like fuck between themelves about whether to accept our trade terms & what demands they insist on. Many of our players will have retired by the time it's sorted I recon.
You've got the right gist, but some of your details are off.
Once we trigger Article 50 (the exit article) we have exactly two years to negotiate our exit. We can extend that period, but only if every single EU member agrees to it, and with the rise of far right parties across Europe demanding their own exit, many countries have a good reason to refuse that extension. The Germans, for example, have indicated that they want us gone as soon as possible. We haven't triggered Article 50 yet, but we will soon - Cameron wants it to be the first act of his successor but the Germans, again, are demanding we do it earlier.
As for Europe accepting our trade terms - it's actually the other way around. There is no doubt at all that the government will request entry into the European Economic Area, meaning access to the free trade and essentially little difference economically to how things were before. However, the EU has made it clear to other EEA members that there are non-negotiable conditions for entry, and they include committing to paying an annual subsidy to fund the EU government and accepting free movement of EU workers - so basically all the things people were voting Leave over. In other words, there is no chance this happens; the general public wouldn't allow it.
Failing that we can negotiate a deal similar to Canada which would get us restricted free trade access but it would be at the liberty of what the EU demands we accept, and since they are furious at us then you can bet we would have to agree to pay subsidies and bend over backwards to EU definitions on what can and can't be sold (fruit having to be within certain weights, products having to be made of certain metal alloys etc). It is also quite unlikely this happens, especially as it's been indicated that the EU will be deliberately vindictive with their terms.
In all likelihood therefore, we are going to have to accept being mostly cut off from the European market, excepting paying extortionate tariffs.
Anyway, to answer the original question, EU players will now need work permits, and while most top players will qualify anyway under the exceptional talent clause, teenagers for the most part won't. The same will happen in reverse, too, so after we leave expect to see Chelsea (and us) not being able to send British kids to European clubs on loan for competitive experience anymore. Also, the pound's value is going to be rocked for a few years while we struggle to rebuild foreign trade links from scratch, so expect the pound to end up anywhere from 20 to 50%, weaker thus inflating transfer fees and deflating them out.