Will Brexit Britain go the same way as the US, with regards to the wholesale export of manufacturing jobs and a massive increase in the trade deficit?

It’s difficult to say because the time-frame for any EU trade deal would be determined as much by politics as economics, however, it is in the financial interests of the UK to keep trading with the EU because the size of the market. The EU might play hardball to discourage other states from leaving the bloc. A deal may also take some time because EU states cannot agree among them how to treat the UK exit. Moreover there is the issue of 50 plus existing trade EU agreements with the rest of the world where the UK participated as a member. The UK would need to renegotiate new trade agreements as an independent entity to carry on trading without facing stiff tariffs and it’s not clear how long this would take.

A bigger issue than the timing of exit is the question of what sort of trade arrangement the Brexit side actually wants. Some seem to want access to the single market in the manner of Norway and Iceland. Nevertheless, economic area membership (EEA) also means accepting the free movement of people into Britain – which is intensely opposed by many Brexit supporters. Others favor a deal with Europe under the World Trade Organization framework. This is how China and the US currently trade with the EU and it would not require any complex negotiations. This arrangement would enable Britain to do away with EU regulation and close its borders to immigration from Europe. But this would also imply that UK exporters will face the EU’s common external tariff.

Almost all depends on what assumption one makes about how post-Brexit trade deal with Europe would be concluded. "Oxford Economics modeled this scenario and assumed a relative 7 per cent fall in trade volumes over 30 years. The London School of Economics’ Centre for Economic Performance calculates that the long-term costs to Britain of lower trade with the EU could be as high as 9.5 per cent of GDP. Leave campaign-supporting economists have not done much detailed modelling. But one study by Patrick Minford of Cardiff University shows a boost to GDP growth by 2020 on the basis of Britain dismantling all tariffs unilaterally post-Brexit. Under Minford’s assumptions this is great boon to some sectors of the economy which benefit from cheaper imports. But as Minford himself admits, this approach implies catastrophic damage to many exporting firms.” 

In this scenario many large companies will prefer to set up shop in the country of interest and manufacture domestically to take advantage of tax incentives and whatever other advantage they can extract from the hosting country for job creation. Of course, this exactly the direction US manufacturing took with respect to China and other countries and exported jobs to countries where labor is cheaper.