I can't find old-enough polls just about immigration from the EU, but on the more general question of immigration: "are there too many immigrants in the UK", the polls have shown numbers that have been consistently high over decades:
This is my own chart based on Ipsos MORI data. I've only plotted the total agree/disagree, leaving out "don't know" and non-responders. The trend lines are 4-degree polynomials, although, i have no theoretical basis for that choice, it just looked good.
Just from this data, it's hard to conclude a clear trend around the time (2004) of the Eastward expansion of the EU. Maybe there was a delayed effect, but that would require more data to ascertain. There does seem to be a general trend toward an in increase in anti-immigrant sentiment in the 2010s compare to the 2000s (or even overall if you plot a straight line); I haven't done any formal testing of this 2000-2010 hypothesis; the conclusion seems sensitive the two low-scoring surveys from the 2000s, but the odd spike in between them makes me skeptical that it is a sizeable real effect. The general trend of a slight increase is less sensitive to removing those two surveys, but the "yes" line is nearly flat; see below
These surveys didn't all have the same methodology; some were phone, some face-to-face, and some by self-completed questionnaires. With such heterogeneity, more data points would be needed for anything but a wild guess in terms of conclusions.