Does the Siege-Gang Commander have haste?
Only creatures in play when Burst of Speed resolves are affected - it even says so in the rulings. But in your example, it won't matter because by the time Goblin Lackey lets you get your extra Goblin out, you're already well past your chance to attack (unless you somehow get a second combat phase).
If you're curious about the actual rules behind this, unfortunately, it's one of the more annoying nitpicky details. Spells and non-static abilities which modify characteristics (name, mana cost, color, color indicator, card type, subtype, supertype, rules text, abilities, power, toughness, loyalty, hand modifier, and life modifier) determine the set of objects they apply to when they resolve. Haste is an ability, which is a characteristic, so Burst of Speed only affects the creatures that are on the battlefield when it resolves.
On the other hand, spells and non-static abilities which don't modify characteristics are considered to be modifying the rules of the game and determine what they apply to continuously. For example, there's Dread Charge, which looks like it's granting something like fear, but since it's not actually granting an ability, will indeed apply to creatures which come into play after you cast it. Alternatively, in the context of your example, if hypothetically haste weren't an ability, and instead Burst of Speed said "Until end of turn, creatures you control can attack even if they haven't been under your control continuously since your most recent turn began", then it would affect subsequent creatures that enter the battlefield.
"Indestructible" was recently "fixed" to become a keyword ability, so like with haste and Burst of Speed, for example, Boros Charm now only applies to creatures in play when you cast it. On the other hand, "is unblockable" was changed to "can't be blocked" to emphasize that it's not an ability, so for example Glaring Spotlight's ability does apply to creatures which come into play after you use it.
Here's the full rule:
Finally, static abilities also apply continuously. This isn't usually confusing; they're written as simple statements of fact, like Blur Sliver's "Sliver creatures you control have haste", which while in play gives all your Slivers haste, plain and simple. The full rule for that:
No, Burst of Speed only affects creatures in play at the time it resolves.
From the ruling on Burst of Speed at the bottom of the page: