Sterling rose to its its highest level since July 29, having as recently as Tuesday plunged to multi-year lows.
Asian stock markets mostly rose, and so did those in Europe except for London as the strong pound hurt multinationals whose earnings are mostly in foreign currencies.
US equity markets were also solidly up in the late New York morning on optimism following news of fresh US-China trade talks, less than a week after the latest round of tit-for-tat tariff measures.
Strong US private-sector hiring data also helped sentiment.
- 'Fresh legs' -
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UK parliamentarians are rushing through legislation designed to keep Johnson from breaking Britain off from the European Union without a trade deal.
They appeared on course to manage this by Monday, only just ahead of a five-week shutdown of parliament Johnson controversially ordered last month.
"News that the bill to stop a no deal Brexit should complete its passage through the House of Lords by Friday afternoon has put fresh legs on the pound's rally," City Index analyst Fiona Cincotta told AFP.
"Whilst there are clearly some hurdles still to be overcome, pound traders are feeling significantly more optimistic about the UK's eventual exit from the EU," she added.
The embattled British currency had tanked on Tuesday to $1.1959, its weakest level since 1985 except for a "flash crash" in 2016.
- 'Not out of woods' -
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"The pound has been behaving as a tidy barometer of the odds of a no-deal Brexit," Rabobank analyst Jane Foley told AFP.
"The pound is clearly not out of the woods with political uncertainty still at very elevated levels -- but the risk of a disorderly Brexit next month at least looks set to be pushed off the table," Foley added.
The bill would force Johnson to seek a three-month Brexit extension until January 31 should no deal emerge from an EU summit in Brussels on October 17-18.
Equities meanwhile were building on the previous day's advance as investors cheered developments on the trade front, in Hong Kong and in Europe.
The leader of Hong Kong on Wednesday withdrew a controversial extradition bill that had sparked months of sometimes violent protests in the financial hub.
News that Italy had formed a new moderate, pro-European government also lifted sentiment and early Thursday China announced it would resume trade talks with the United States in Washington next month.
- Key figures around 1540 GMT -
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Pound/dollar: UP at $1.2328 from $1.2253 at 2100 GMT
Euro/pound: DOWN at 89.63 pence from 90.06 pence
Euro/dollar: UP at $1.1046 from $1.1035
Dollar/yen: UP at 107.00 yen from 106.39 yen
London - FTSE 100: DOWN 0.6 percent at 7,271.17 points (close)
Frankfurt - DAX 30: UP 0.9 percent at 12,126.78 (close)
Paris - CAC 40: UP 1.1 percent at 5,593.37 (close)
EURO STOXX 50: UP 1.0 percent at 3,484.70
New York - Dow: UP 1.7 percent at 26,789.89
Tokyo - Nikkei 225: UP 2.1 percent at 21,085.94 (close)
Hong Kong - Hang Seng: FLAT at 26,515.53 (close)
Shanghai - Composite: UP 1.0 percent at 2,985.86 (close)
Brent North Sea crude: UP $1.50 at $62.20 per barrel
West Texas Intermediate: UP $1.25 at $57.51