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In a Fox News broadcast (timestamp 2:59), Greg Palkot (London-based senior foreign affairs correspondent for Fox News) said of Julian Assange:

"He has cost the UK government and security forces, police here, millions and millions of dollars. They've been guarding that place, waiting for him to come out, to snag him for the past several years..."

Did the UK government really spend millions of dollars to keep the Ecuadorian Embassy guarded for several years to prevent Julian Assange from leaving it and escaping?

How much did it cost and what was the money spent on?

Yes, back in 2015 a figure of around £11 million was estimated by the Metropolitan Police. This figure represents the total amount of resources allocated to monitoring the embassy for the three years between Assange entering the embassy (June 2012) to April 2015.

From The Telegraph in June 2015:

The embassy, behind Harrods in Knightsbridge, is watched by police stationed on the corners of the building, and an officer inside the foyer of the multipurpose red brick residence at all times. The Metropolitan Police refused to discuss how many policemen were deployed to the embassy, but they did confirm the cost.

"As with all long term operations, issues around resourcing are subject to regular review in an attempt to minimise costs," a spokesman told The Telegraph.

"The estimated full cost to April would be £11.1m. The costs provided are an estimate based on averages, as actual salary and overtime costs will vary daily."

The Met said the figure included £6.5m of what they termed "opportunity costs" – police officer pay costs that would be incurred in normal duties – and £2.7m of additional costs such as police overtime. A further £1.1m was put down to "indirect costs" such as administration.

Around October 2015 the constant monitoring and police presence was removed due to being "no longer proportionate", so presumably the figure today is not much higher than it was in 2015.

A bit of context for these numbers and timelines can be found at this related Politics SE question.

No additional funds were spent on his capture beyond the spending already budgeted for 'Diplomatic Protection'.

The Diplomatic Protection Group is responsible for the guarding of embassies. The money that was spent trying to apprehend Julian Assange from inside the Ecuadorian Embassy (or rather, waiting for him to come out) was covered by the existing budget assigned by the Metropolitan Police for the protection and guarding of embassies in that part of the UK.

The Metropolitan Police said the costs were covered by the budget for diplomatic protection, which provides policing for embassies in the UK.

_{^{BBC News: Julian Assange: Costs of policing Wikileaks founder reach £10m}}

Since that money was already budgeted for the guarding of embassies, since that money was spent on guarding an embassy and since no additional funds were subsequently added to their budget to cover the shortfall caused by guarding that embassy, it's arguable that no additional money was spent on his capture.

That being said, following the same argument to its logical conclusion, it did result in a focusing of the DPG's resources on a single embassy instead of all of the foreign embassies in the area.

According to the BBC, the cost was over £10 Million as of February 2015.

Per this BBC report from 6 February 2015

Scotland Yard has spent about £10m providing a 24-hour guard at the Ecuadorean embassy in London since Wikileaks founder Julian Assange claimed asylum there, figures show.

...

Between June 2012 and October 2014, direct policing costs were £7.3m, with £1.8m spent on overtime, police said.

Scotland Yard confirmed the cost of the operation to UK taxpayers in the first 28 months, until 31 October last year, had reached £9m.

...

The cost of a further three months policing is now expected to have taken the total bill to about £10m.

The figures - which equate to more than £10,000 a day - were obtained by LBC radio under the Freedom of Information Act.

This obviously does not contain the total amount spent, as these numbers were current as of early 2015, but is well within the range of "millions and millions of dollars".

Just to bring the information in the other answers together ...

In the UK the phrase "UK Government" is taken to mean the Prime-minister and their cabinet members (including the home secretary responsible for policing in England and Wales - but not in Scotland or Northern Ireland).

The police forces, not politicians, choose how to allocate their resources to individual investigations. In operational matters the UK police forces are independent of government

These distinctions are very important in the UK.

Did the UK government pay “millions and millions of dollars” to try to snag Julian Assange?

No, as I understand it, the UK Government did not allocate any funds specifically for this purpose.

Yes, from 2012 until 2015 the Metropolitan police chose to spend several million pounds of their policing budget, attempting to execute an arrest warrant on Assange. After 2015 they spent very little, relatively speaking.

Note that neither the UK government nor the Metropolitan police issue arrest warrants (AFAIK that is the job of the judiciary, it is they who wished to snag Mr Assange)