Some of my advice is the exact opposite of Ankur's!
I've lost my VISA card three times travelling, all three times in countries without their own VISA offices: Vietnam circa 2000, Nicaragua circa 2007, and Georgia in 2012. I only took advantage of the "emergency cash" assistance on the second occasion.
I've also met travellers with lost, stolen, or blocked cards both overseas when I've been a traveller, and at home in Australia where I work in a backpacker hostel. The last time was just today in Bangkok!
One reason VISA offers a USA-based "universal" emergency number is because when you can't call a local VISA office it's a useful backup. It turns out that at least some US 800 numbers work internationally, unlike some other toll-free numbers in other countries. It probably will incur a toll when called internationally, but should still work.
In Vietnam the first time I went to a telephone & Internet place. First I used the Internet and Googled three VISA office phone numbers in nearby countries. I was a bit pessimistic that I'd be lucky on the first number, and this turned out to be the case. The VISA people were very professional and efficient and spoke quickly and coherently knowing I was paying for a long-distance call. They cancelled my card before anybody who stole or found it had a chance to use it. I didn't need emergency cash this time.
In Nicaragua the second time I borrowed the Internet and telephone in a bookshop and tried same procedure as last time. This time I couldn't manage to find a way to call any of the numbers toll-free or reverse charges. I ended up calling the VISA emergency number in Australia after several false starts but then had to reimburse the bookshop for what was probably an expensive call with no real way to know exactly how much the call would cost.
Again the VISA staff were efficient and professional and did not waste unnecessary time. This time I needed both an emergency VISA card and emergency cash as I had had my whole wallet stolen, not just the VISA card as previously.
I was able to get emergency cash on the same day if I could make it to a Western Union branch before closing time. The branch was reasonably close to my hostel and I managed to jog there and get cash to pay back the bookshop and pay my hostel and living costs while waiting a few more days for my emergency card.
In Georgia the third time I ended up using Skype and can't recall if I called VISA or my bank. I didn't need emergency cash but did need an emergency card. This time the bank would only send the two cards (VISA and ATM) by courrier, separately, at a cost of $50 each! This annoyed the hell out of me so I didn't get the replacement cards sent. Luckily it turned out a friend of a friend was soon going to Australia and returning to Georgia and he brought me my cards, saving the $100.
Today in Bangkok a Canadian guy at my hostel called the VISA emergency number in Thailand, which does have its own VISA office - but was told the emergency money would take 24 hours! He's flying tonight so had to find another way to get some emergency money.
Various other times I have met people who cancelled lost or stolen VISA (or MasterCard) cards via their bank rather than directly via the card companies' emergency hotlines. Several of these had ended up as nightmares of incompetence taking months before the people had access to their funds again! For this reason I advise everybody in this predicament to report their missing cards to VISA if possible before bothering to contact their bank. The people on the VISA hotline are experts who deal with lost and stolen cards in every combination of citizen X in country Y every day. People working in banks mostly do other stuff and can mess up when your card goes missing in some exotic locale.
TL;DR
Call the VISA lost/stolen card hotline before your bank unless you have a special relationship with your bank and trust them not to make things worse before they make them better.
I've personally got emergency money on the same day in an exotic location. But I've personally tried to help somebody in a similar predicament in a less exotic location where they couldn't get the money on the same day.
I don't know why I didn't have to wait and he did. It could be extra security against fraud and identify theft over the past dozen years. It could be due to arcane rules and regulations beyond VISA's control that differs from country to country.
