Did Lenin order the execution of the Romanovs?

There was a strict party hierarchy and Goloschokin (head of Ural soviets) answered directly to Sverdlov, who in turn answered to Lenin. It’s unlikely such a major decision would have been made without consent from the top, and the ural Soviets would not have dared to act independently.

On April 6 Sverdlov (Lenin’s right Hand man) told Goloschokin that Yakovlev was being sent from Moscow to transfer Nicolas to Ekaterinburg instead of Moscow, which indicates they no longer planned to put him on trial.

Goloschokin, head of the Ural Soviets, traveled to Moscow on June 30th to meet with Lenin and Sverdlov to discuss the Romanov fate; he met with them on July 3 where it was decided that the Ural Soviets should ready the details and once military situation dictated it, contact Moscow for final approval for execution.

Trotsky stated that Lenin and Sverdlov had decided there should be no living banner left for the Whites to hang onto.

Yurovsky insisted that the decree for the go ahead was given to him on the 17th by Goloschokin, which was most likely the previously drafted decree brought back from Moscow as there was no telegram received that night from Moscow.

However, a Kremlin guard later stated that a telegram from Moscow was sent that night to Ekaterinburg and the evidence destroyed.

As for the years of denials by the Soviets that Lenin was responsible, of course they would want to protect him from being associated with such an unsavory episode.

From Robert Messie "Nicholas and Alexandra: The Classic Account of the Fall of the Romanov Dynasty." (If you do not know, Ilyich in this context means Lenin.)

The link between the party leaders in Moscow who authorized the murder and the Ural Soviet which determined the time and method of execution was later described by Trotsky. He explained that he had proposed a public trial to be broadcast by radio throughout the country, but before anything could come of it, he had to leave for the front.

“My next visit to Moscow took place after the fall of Ekaterinburg. Talking to Sverdlov, I asked in passing: ‘Oh, yes, and where is the Tsar?’

‘It’s all over,’ he answered. ‘He has been shot.’

‘And where is the family?

‘And the family along with him.’

‘All of them?’ I asked, apparently with a touch of surprise.

‘All of them,’ replied Sverdlov. ‘What about it?’ He was waiting to see my reaction, I made no reply.

‘And who made the decision?’ I asked.

‘We decided it here. Ilyich believed that we shouldn’t leave the Whites a live banner to rally around, especially under the present difficult circumstances.’

I did not ask any further questions and considered the matter closed. Actually, the decision was not only expedient but necessary. The severity of this summary justice showed the world that we would continue to fight on mercilessly, stopping at nothing. The execution of the Tsar’s family was needed not only in order to frighten, horrify, and dishearten the enemy, but also in order to shake up our own ranks to show that there was no turning back, that ahead lay either complete victory or complete ruin.… This Lenin sensed well."

Edit. Here is more from the same book:

on July 20 (of 1918), the official proclamation mentioned only Nicholas.

“DECISION of the Presidium of the Divisional Council of Deputies of “Workmen, Peasants, and Red Guards of the Urals": In view of the fact that Czechoslovakian bands are threatening the Red Capital of the Urals, Ekaterinburg; that the crowned executioner may escape from the tribunal of the people (a White Guard Plot to carry off the whole Imperial family has just been discovered) the Presidium of the Divisional Committee in pursuance of the will of the people, has decided that the ex-Tsar Nicholas Romanov, guilty before the people of innumerable bloody crimes, shall be shot. The decision of the Presidium of the Divisional Council was carried into execution on the Night of July 16th–17th. Romanov’s family has been transferred from Ekaterinburg to a place of greater safety.”

...

A year later, unable to maintain their fiction, the Bolsheviks admitted that the entire family was dead. They still did not admit their own responsibility for the murders. Instead, they arrested and brought to trial twenty-eight people, all Social Revolutionaries, who, it was charged, had murdered the Tsar in order to discredit the Bolsheviks. Five of the defendants were executed. The hypocrisy of this second crime was later admitted by the Bolsheviks themselves in Bykov’s book.

[1] Trotsky, Leon, The History of the Russian Revolution. 3 vols. Translated by Max Eastman. New York, Simon and Schuster, 1932.

[2] Bykov, P. M. The Last Days of Tsardom. London, Martin Lawrence, 1934.

P.S. The story with the initial official denial of the execution of the rest of the family an then blaming PSR members, I find a bit strange, more precisely, out of character for the Bolshevik leadership: They were not shy (actually, quite proud of) of admitting mass terror against the "class enemies". Executing spouses and children of the latter was quite consistent with the (theory and practice of) Communist morality. However, it is irrelevant regarding the original question.

tl;dr Nobody knows.

If you haven't found anything conclusive, you are not alone: there's no agreement among the historians, either. Some say Lenin was not at all responsible, that the execution was sanctioned by the local government and that Lenin and the top party members were informed about that post factum. Some say Lenin was the one who gave the orders. And there's a whole range of opinions in-between.

While there are data sources to support both versions, they are all too vague, ambiguous and contradictory to reliably prove or disprove any of them. Take Trotsky's quote from the answer of Moishe Cohen, for example. Quite clear, isn't it? Yet, Trotsky himself gives a different account of the situation in other sources; according to the records, Trotsky was also present at a meeting on July 18, where Sverdlov announced the execution (Sverdlov himself got a telegram about it on 17th, and on 19th it was already in the newspapers), so the whole questioning of Sverdlov and "surprise" of Trotsky doesn't make much sense. There’s no direct evidence (as in a written execution order signed “Lenin”), but then again, some argue that doesn’t mean much, because at the time orders were often given orally, without paperwork.

The version that Lenin was not involved isn't that "odd" if you understand that at the time Russian Bolsheviks weren't yet that all-powerful united totalitarian force they later became. Not only did they have to share power with other parties, the Bolsheviks themselves weren't a monolithic entity. Lenin had never got anywhere near the level of personal dictatorship Stalin later achieved: his orders were quite often disputed or even openly sabotaged by his Party comrades. Lastly, the local Soviets, although accountable to the Central Executive Committee, had a lot of autonomy, especially since communication was often problematic, with the civil war and all.

At the time of their execution, the Romanovs were at Yekaterinburg, the capital of the Ural region under control of the Ural Soviet (they were moved there instead of Moscow as a result of a conflict between the Ural Soviet and Moscow, their Moscow-appointed escort had to actually protect them from attempts on their lives, you can read about that, for example, in “The Russian Revolution” by R. Pipes; Pipes, by the way, states many times that the execution orders came from Lenin, yet mentions no evidence, except maybe the Trotsky quote cited above). The Ural Soviet was dominated by the SRs, anarchists and left-wing Bolsheviks (even more radical, believe it or not, than the "mainstream", Lenin-style Bolsheviks, who were stronger in Moscow and St.-Petersburg). They wanted to execute the Romanovs for several reasons: execution of "the crowned murderer" was a popular demand even before the Bolsheviks came to power; they had to show their revolutionary radicalism; they, possibly, wanted to sabotage the Treaty of Brest-Litovsk, which they were opposed to. The Ural Soviet sent their representative to Moscow several times, asking to authorize the execution. According to the official version, the authorization was never given: the Bolshevik leaders didn’t want to execute the Romanovs [yet], they planned to put them on trial some time later (at the time they had more immediate concerns, with the civil war and all). There are evidences that they planned to use the Romanovs in re-negotiation of the peace terms with Germany (as they were relatives of the German Imperial family; for that, obviously, they were needed alive; that was also one of the reasons the Bolsheviks were denying the execution later). When the White Armies came close to Yekaterinburg, the Romanovs were executed by the Ural Soviet without a trial to prevent them from being liberated by the Whites. Some argue that the Ural Soviet wouldn’t dare do that without Moscow’s consent; the fact that no one was punished by Moscow for that speaks in favor of that version; on the other hand, the reason may have been that Moscow did not want to intensify their conflict with Yekaterinburg or admit how little power they had over the Ural Soviet at the time.

Most historians agree that Nicholas himself would eventually have been executed even if a trial had taken place; his wife and children probably wouldn't. The Bolshevik's terror against the "class enemies", although brutal, wasn't that consistent, many of the said "enemies" were allowed to emigrate, some stayed and served under the Bolshevik's government (of lesser caliber than grand princes and arch-dukes, obviously). No good in turning enemies into martyrs - Prince Gabriel Romanov was released from prison by Lenin (Lenin!) for that reason (his relatives were kept in prison and executed later, though). There was also a somewhat similar story with other members of the Romanov family, of Grand Duke Alexander Mikhailovich: they were kept in the Crimea. The local Yalta Soviet pressed for their execution, but their guards didn’t allow it without authorization from Moscow, despite advancing German troops. Either the guards were more persistent, or those Romanovs were luckier than their crowned relatives, but they kept their lives, were liberated by the Whites and emigrated.

UPD.: One important thing I missed is that the situation with capital punishment in Russia at the time was complicated. It was abolished by the Provisional government in February 1917, restored soon, but only in the army for military crimes, abolished again by the Soviet government on October 28, 1917, and restored on June 13, 1918 (one month before the Romanovs execution). Did that somehow influence either the Ural Soviet or Moscow? E. g. the Ural Soviet (not knowing in advance how the law would change) didn't want to give the Romanovs to Moscow for their open trial, because in that case they could not be executed? Was Moscow postponing the trial indefinitely for that reason as well?

UPD.: Some sources, what I could find in English.

Supporting the version that Lenin was involved:

R. Pipes, The Russian Revolution.

S. Melgunov, Fate of Emperor Nicholas II after the Abdication.

Supporting the version that Lenin was not involved:

P. M. Bykov, The Last Days of Tsardom. (there's also a book of the same author on Amazon called "The Last Days of Tsar Nicholas", they both may be different translations/editions of his Russian book "Последние дни Романовых", but I am not sure).

For those who read Russian:

Г. Иоффе, Революция и судьба Романовых (couldn't find any information about English translation, maybe it doesn't exist - too bad, it looks quite unbiased).

There was also a recent official investigation by Soloviev (В. Н. Соловьев), some of his interviews can be found in Russian internet (this one, for example - don't know if Google Translate would make it understandable).

The Russian wikipedia page on the matter is quite good, many more sources (I think everything listed by me or Moishe Cohen is present there as well), but, like I said earlier, too numerous, fragmentary and Russian.