A high intensity proton beam acceleration with a continuous wave output of roughly 1mA producing 1GeV protons, which seems reasonable to me, will accelerate a mole of protons every 96.5 million seconds.
That is pretty much 3 years per mole.
If we assume every proton impacts a 238U nucleus in the 'primary' target and undergoes ludicrous spallation-fission with the typical 20 daughter fast neutrons produced that gives us a mole of neutrons every 54 days.
If we assume no downstream fissions (which is probably unrealistic but lets go with it as a conservative estimate) and all of the neutrons are absorbed in a solid target compposed entirely of something with atomic mass ~238 then we would expect to produce roughly 4.3g of fissile material per day.
If they all fission and double the number of neutrons that improves but still looking at grammes per day.
Which is enough for it to be a worry for a bomb programme (as linacs get much cheaper) but not really for civil use.
EDIT:
Correction, this
study seems to suggest a surprisingly efficient LINAC with a 120mA beam current at up to 1GeV is feasible, with ~57% efficiency at high currents. And this is
CERN so they tend to know what they are talking about for accelerators.
120mA beam current with the x20 multiplication assumption tends to lead to a mole of accelerated protons every 223hrs, which leads to a mole of daughter neutrons every 9.3 hours, which leads to roughly ~600g of fissile material produced per day.
Assuming you use uranium for the primary target and thorium for the primary target you might bea ble to produce enough fissiles to power the accelerator.
After all 600g/day is 219kg per year, which diluted to 4.2% for an ESBWR (assuming 233U is the same as 235U, which I know it isn't) with 0% DU is ~5.2t of fuel, which has an electricity production of ~2TWh (e) for an average power of ~230MWe, whereas the accelerator would require something like 175MWe [drive power, not beam power].
So you are actually surprisingly close to breakeven, even diluting the 233U with pure 238U rather than NU or RepU. Using ~0.7% enriched U (either NU or RepU) would give us 6.2t of fuel per year, raising average reactor power to ~300MWe, giving double the beam input power.
Sure its still rubbish compared to more conventional technologies, but honestly I am amazed it actually gets near breakeven at all.