“Any restriction on travel is going to be awkward at the least.”
I don’t know if you remember (or have researched) the situation before the UK was an EU member. People travelled to mainland Europe with no more difficulty than they do now. They had to show a passport, that’s all. No visa required, no advance permission required. You got on your boat or plane, flashed your passport on arrival and that was that. Companies such as those you describe who had branches in Europe and wanted their staff to work there had a simple permit to apply for which lasted (if my memory serves me correctly) three years. Not really awkward, easily achievable before things like that were dealt with online so probably very much easier now. In short, the problem you foresee (like many you seem to be troubled about when Brexit is mentioned) is a non-problem.
Set this minor inconvenience (which may not even exist at all, depending on what arrangements are agreed) against the advantages of withdrawal and it is extremely small beer indeed. Life, business and commerce goes on despite the best efforts of politicians to place obstructions in the way. Cambridge Science Parks (and similar institutions) will fare perfectly well. Any minor inconveniences which may or may not be introduced will be easily overcome. My nephew’s wife works for an international concern that has facilities in France, USA, Canada and Dubai as well as the UK. She works from time to time in all of them and has no more difficulty working in the non-EU countries as she does in France. Brexit is far too great an opportunity to miss because a few scientist in Cambridge are unwilling or unable to make an online application to work elsewhere.
Yes I do realise a vaguely similar situation existed in Ireland, Peter. But I would say that the two situations (and these proposals) are not that similar. But if the Euromaniacs are daft enough to offer people from non-member nations the same rights as those from member nations (principally, it seems, to feed their vanity and to protect the status of their vainglorious "European Project") then good luck to them. So long as the concession is not reciprocated by the UK government, that is.
As an aside, I fail to see why people would need to "register" for the concession. Surely all that is needed is to grant it to anybody with a UK passport. Unless, of course, they'd like to demonstrate how many people are "proud" to have associate membership of the EU.