Now that the UK has voted to leave the European Union, how is that going to affect the European Space Agency? How much funding and participation does ESA receive from the UK based on the EU relationship? Are there other potential issues with the UK leaving the EU in regards to ESA?
The EU and ESA are not related. They have different member states, where neither is a subset of the other. The only potential issue is simply economic. If the British exit from the EU results in a depressed UK economy, then they may elect to participate less in ESA.
The effect should be small, but possibly non-zero. There currently exists growing cooperation between EU and ESA.
Within the EU-ESA membership agreement, one of the noted obstacles is the membership asymmetry:
2.2. Membership asymmetry
At present, 17 EU Member States are Members of ESA. ESA's members include Norway and Switzerland, which are not EU members. Canada has a bilateral cooperation agreement with ESA. As the collaboration between the EU and ESA grows, this asymmetry combined with a voting system where each Member State has one vote in the ESA Council and the key decisions within this body are adopted by unanimity gives ESA members, which are not members of the EU, disproportionate leverage over matters that may affect the EU. In addition the asymmetry complicates discussions particularly as regards security and defence matters as indicated below.
If the UK leaves the EU, the UK would be added to the ranks of Norway and Switzerland. Currently, the UK influences ESA directly as a member state, and indirectly through the EU-ESA cooperation. Its influence would likely diminish if it becomes an ESA-only member like Norway and Switzerland.
This UK House of Commons publication contains more information.
On paper it should make no difference given that there are many non EU members contributing to the ESA budget. Reality, however, is often very different than paper.
The politics of the current situation are difficult to predict. Certainly, the national governments of France, Germany and Italy will find it politically impossible to support any major collaborative aerospace projects based primarily in the UK.
If the UK feels marginalised by the major EU powers within ESA then the UK government may respond by reducing the contribution to the budget, which will exacerbate the problem.
I don't think its possible to extrapolate any further given the current global political and economic uncertainty following on from last Thursday's referendum result.