* Sterling down to lowest level vs dollar since mid-1985
* Pound's weakness gives lift to UK stocks
* FTSE 100 closes up near 2015 record highs
* FTSE 250 mid-cap index touches new record highs
* Pound's slump has hit dollar value of UK shares (Adds fund manager quote, details)
By Anirban Nag and Sudip Kar-Gupta
LONDON, Oct 4 (Reuters) - Sterling slid to its lowest level in more than three decades on Tuesday on fears of a "hard Brexit" from the European Union and its single market, although the weaker pound sent UK stocks surging higher.
The pound has already lost 1.7 percent against the U.S. dollar since Prime Minister Theresa May said on Sunday the formal process to take Britain out of the EU would start by the end of March 2017.
On Tuesday, she added the divorce from the EU will not be "plain sailing" and there would be "bumps in the road".
The pound fell to $1.2735, its weakest since mid-1985. Earlier, sterling also hit a three-year low of 87.65 pence per euro and a 6-1/2 year low on a trade-weighted basis.
Many economists and investors fear May's government will back a "hard Brexit" option where Britain quits the single market in favour of imposing controls on immigration.
That could hinder inward and outward trade and constrict the foreign investment needed to fund Britain's huge current account deficit, one of the biggest in the developed world.
Economic activity has held up better than many had expected since Britons voted in a June referendum to leave the EU, but many policymakers are anxious about the prospects for future investment. The Bank of England launched a stimulus package and cut rates to record lows in August and may ease policy again in coming months, which could drag the pound lower.
"Most of the key (BoE) members have expressed a willingness to continue acting pre-emptively ... and an expectation that more easing is likely to be necessary," UBS strategist John Wraith said.
"Additional stimulus would likely drive further sterling weakness," he added, reiterating the bank's forecast for $1.20 per pound and parity with the euro by end-2017.
UK stocks benefited from the pound's weakness.
The blue-chip FTSE 100 index, dominated by international and export-driven companies that often benefit from a weaker pound, closed up 1.3 percent at 7,074 points - close to a record high of 7,122.74 points set in April 2015.
The fall in sterling has given a boost to many of the FTSE 100's international companies which earn much of their revenues in U.S. dollars, and therefore get a currency-related accounting lift as those dollars are converted back to pounds.
However, the slump in sterling has also impacted the U.S. dollar value of FTSE 100 stocks, a potential negative for overseas investors for whom the dollar is their benchmark currency reference.
"Investors may feel optimistic now that the FTSE 100 has broken through the 7,000 barrier, but we might not want to crack open the Champagne yet," said Nick Peters, multi-asset portfolio manager at Fidelity International.
"The market has primarily been boosted by the sharp depreciation in sterling post-Brexit, with the pound having fallen by around 13 percent from its pre-referendum highs. Around 75 percent of the earnings from FTSE 100 companies come from outside the UK, so sterling depreciation effectively makes these earnings worth more," added Peters.
The FTSE 250 mid-cap index, whose companies are more exposed to the state of the domestic UK economy, also closed 0.9 percent higher, having touched a record high earlier in the day.
The FTSE 250 index has gained around 24 percent since the lows struck just after the referendum result on June 24.
The mid-caps have benefited from recent upbeat data and surveys of British households and businesses that have led many forecasters to drop predictions that the British economy will slip into recession this year.
Paul Spencer, portfolio manager at Franklin Templeton's UK equity team, added the slump in sterling also meant UK stocks had become cheaper for overseas investors, and also more affordable for takeover from foreign companies.
Earlier this year, technology group ARM agreed to be bought by Japan's Softbank, while SVG Capital has also had bid interest, with such deals helping lift UK share prices.
"The same value assessments that make UK mid-cap stocks attractive to overseas investors could, we think, lead to heightened levels of mergers and acquisitions in this space in the coming year," said Spencer. (Writing by Anirban Nag; Editing by Janet Lawrence)