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Stuart Agnew is a Member of the European Parliament for the Eastern Counties (in the UK) and stood on the UKIP ticket. His web site describes attending a global warming conference in the "mid-Eighties" at the University of East Anglia. He writes:

The highlight of the conference was a series of maps of our Norfolk coastline. The first map displayed the status quo, the second what it would look like if sea levels rose by one metre, the third five metres and the fourth ten metres. We were assured that, within 30 years, sea levels would have risen by at least one metre, probably five metres and possibly ten.

Were any climate scientists (meaning people who might reasonably have been invited to give a talk at a scientific conference) in the 1980s predicting a 1 meter sea level rise within 30 years? If so, were these predictions part of the mainstream consensus within climate science at the time?

The 24 October 1983 report Projecting Future Sea Level Rise: Methodology, Estimates to the Year 2100 , 2nd edition, predicted (mid-range scenario, see table 4.1 on page 39):

by 2000:

8.8-13.2 cm

by 2025:

26.2-39.3 cm

The mid-range scenario assumed a climate sensitivity of 3.0 degrees C per doubling of CO2 concentration. The mid-range scenario was further split into two scenarios, one where rise due to net melting equaled rise due to thermal expansion, and a second where rise due to melting was twice that due to thermal expansion.

In addition to the mid-range scenarios, extreme low and high scenarios, were considered.

In the low scenario assumed was: climate sensitivity of 1.5 degrees per doubling of CO2 concentration, partial mitigation of CO2 increase due to halving the price of nuclear energy, low heat diffusivity of the ocean, low rates of increase of methane, N2O, and CFCs, low economic productivity growth, rise due to net melting equaled rise due to thermal expansion, and a constant 53% of CO2 emissions being retained in the atmosphere.

The low scenario yielded an estimated rise of 4.8cm by 2000 and 13.0cm by 2025.

In the high scenario assumed was: climate sensitivity of 4.5 degrees per doubling of CO2 concentration, high heat diffusivity of the ocean, high rates of increase of methane, N2O, and CFCs, high economic productivity growth, rise due to net melting being double rise due to thermal expansion, and an initial 60% of CO2 emissions rising to 80% being retained in the atmosphere.

The high scenario yielded an estimated rise of 17.1cm by 2000 and 54.9cm by 2025.

Actual change to 2000 was about 3cm and to present about 8cm.

So in summary, all the scenarios resulted in predictions of sea level rise of well below 1 meter by the present (2018) time. Actual rise has been slightly below the lowest of the scenarios.

The conference would have presumably been organized by the Climate Research Unit. They have published proceedings for similar conferences going back to at least 1973. It would be ideal to track down the relevant proceedings to identify the exact meeting and paper to which the quote is referring, but this would require some library research.

How about other sources on the science in general as of the mid-1980s? Here is a report from 1983 which accepts a projection (on p. 2) of a 10 cm sea level rise over 25 years. And here is a journal article from 1987 which states (on p. 17):

Available estimates generally imply a rise on the order of one meter
in the next century.

This is much less than one meter in 30 years. So clearly, while Agnew's statement may or may not be an accurate description of one particular research presentation about Norfolk, it does not represent the scientific consensus of the mid-1980s for global sea level rise projections.

To add to DavePhD's answer, the High range prediction was 17.1cm in 2000 and 54.9 cm in 2025.

Also, from the page following the one he cites:

AN ALTERNATIVE METHOD PRODUCES SIMILAR ESTIMATES OF SEA LEVEL RISE

To cross check our projections, we estimated sea level changes by another
method: extrapolating past associations between temperature and sea
level. Sea level rise in the last century has been estimated at 10 to
15 cm (4 to 6 inches). The surface temperature rise for the same period
has been estimated at 0.4C. Thus, the ratio of sea level rise to
temperature is somewhere between 25 and 37 cm for each degree.
Including the effects of trace gases, global warming should be
equivalent to at least a quadrupling of CO2 by 2100, which would
raise surface air temperatures by 3.0C to 9.0C (based on the National
Academy of Sciences' range for climate sensitivity, ignoring delays
caused by the heat absorbing-capacities of the oceans). using the 25
cm to 27 cm ratio for the 3C to 9C range yields sea level rises of 75
cm to 333 cm. These estimates are in line with those produced by our
more elaborate approach.

If I'm reading correctly, that 75-333cm range is for 2100, but I'm not sure. This may be the source of the quote, though.

I emailed Stuart Agnew to ask for more information. He forwarded my email to Rev. Philip Foster, who has provided this piece of evidence from "People" magazine on 8th October 1979. It features Dr Gordon MacDonald apparently predicting a sea level rise of 2 meters by 2030.

However what Dr MacDonald actually said in his Congressional Testimony is considerably tamer than this headline (see page 101 in the linked document). After emphasizing how little was known about the way in which sea temperatures would affect the West Antarctic Ice Sheet (WAIS), he suggests that once the sea temperature rose the WAIS could then shed 1/3 to 1/2 its mass over 100 years (my emphasis), and that complete melting of the WAIS would cause a 5 meter sea level rise. Presumably this is where the headline writer got the "up to here" bit. Nothing in Dr MacDonald's testimony suggests that this would happen by 2030.

Update

DavePhd has located the original People article on line. The article describes a "disaster movie" scenario "early in the 21st century" where increasing CO2 leads to temperature rises of up to 20 degrees Fahrenheit and a 20 foot sea level rise. However the article is ambiguous: its not clear whether "early in the 21st century" refers to just the CO2 increase or to the effects.

At the end of the article there is also a quote about world catastrophe "not 200 years from now but within our lifetime". Again the article is unclear, but this seems to have been a quote from an Energy Committee staffer rather than MacDonald.

Update 2

I emailed University of East Anglia to see if they had any records of Stuart Agnew attending a conference. They took it as an Freedom Of Information Act request but they didn't have any records back that far. Not surprising, but it was worth a try.