Take a strong electric field like 10E6 V/m. The size of a uranium nucleus is around 15E-15 m. The voltage drop over the nucleus will be less than 1.5E-7 V, probably much less because the electron cloud will shield the nucleus from most of the electric field.
Just an example of what might be needed which is not inconsistent with this: a slight distorting of the electron cloud surrounding the nucleus might alter the screening a small but appreciable amount. The Gamow barrier calculation is extremely sensitive to minute changes in barrier width, so not much of a change would be required. There could also be an outsize effect from transient asymmetries in this screening across the nuclear volume.