I'd guess that Europe increasing military budgets will be a gradual process (there isn't the money for an instant massive increase). So Governments might decide it's better for them to develop arms manufacturers within their own country (so the retain some of the increased expenditure and reap more of the benefits like employment, taxation, etc.) or at least within the EU.
And developiong local arms manufacturing can be done at the same time as budgest gradually increase.
Ian
To be an effective, independent force, would cost an incredible amount of money. The US, with all of it size, spends getting on for 1 trillion dollars on its military, each year. Europe, doesn't need 10 new aircraft super carriers every 20 years but even so, to catch up with a fully independent defence industry and military would require a significant rework and massive investment of the major economies in Europe. Most a running high debt, national health services and high social welfare systems. The social and health welfare budgets would come under increased threat of real cuts to pay for it. There will be little appetite for increasing taxes and will be the death of any Government attempting to impose it, especially in the UK if it will be perceived as defending the EU, i can see Farage's mouth moving to those words now...
Developing local arms manufacturing is only part of the story, there are large defence companies in europe, such as BAE and Rheinmetall, many are involved in shady stuff like bribery and corruption but glossing over that, many have large contracts with the US defence department and are reliant on that. Still, there are hopes of some independent stuff. The Tempest gen 6 fighter is an example involving the UK, Italy and Japan but then there lies another problem, France and Germany are going there own separate way on their gen 6 fighter so fragmentation is also screwing things up and if Europe does find itself in mortal danger with an invasion ongoing, you've got multiple supply lines needing to be maintained with different factories and countries scrambling for the same limited resources...
There needs to be an agreed consensus, Im sure that could be achieved within the EU where naturally, our voices will be heard. Oh, we left didn't we?