Continue a loop in Julia when the time out

Simplest solution maght be to use the maxiters flag in solve(... ;maxiters=1000, ...). If you realy need a more more accurate timing you can use a callback function:

using DiffEqBase, OrdinaryDiffEq

u0 = [1.,0.]
function fun(du,u,p,t)
   du[2] = -u[1]
   du[1] = u[2]
   sleep(0.1)  # throttle the function
end
tspan = (0.0,10.0)
prob = ODEProblem(fun,u0,tspan)


struct Timer
    tstart::Array{Float64,1}
    runtime::Float64
    Timer(t) = new([0.0],t)

end

function (timer::Timer)(u,t,integrator) 
    if timer.tstart[1] == 0.0 
                timer.tstart[1] = time()
    end
    time() - timer.tstart[1] >= timer.runtime && return true
end

affect!(integrator) = terminate!(integrator)
cb = DiscreteCallback(Timer(60),affect!)
sol = solve(prob,Tsit5(),callback=cb)