Considering the Lord's Supper was instituted on the Passover, would the apostles have concluded it was to be an annual celebration like the Passover? When instituting the Lord's Supper, Jesus commands the apostles: "Do this in remembrance of me."

Luke 22:19 reads:


  Then he took bread, and after giving thanks he broke it and gave it to
  them, saying, “This is my body which is given for you. Do this in
  remembrance of me.”  (NET)


As the Passover was a yearly celebration and the Lord's Supper was instituted during it (apparently to be part of the Passover meal or even a replacement for it), it would seem natural for the apostles to understand the command "Do this" to mean do this annually during or instead of the Passover meal. Additionally, all the major festivals of the Jews were annual. 

However, in 1 Corinthians 11 it seems that the Lord's Supper was not merely an annual celebration and distinct from the Passover meal. 

Considering the occasion when the Lord's Supper was instituted, why would the early Church not conclude it was an annual celebration, and that it could (or was intended to) be divorced from the Passover meal? 

Although there are accounts of the resurrected Jesus eating with his disciples, they are not explicitly celebrations of the Lord's Supper.
I think that the answer lies in how the Church understood the nature of the Eucharist.  Whereas Passover was essentially a commemoration of an event of the Exodus, the Eucharist is something entirely different.  According to the account in Exodus (12:14), Passover was instituted strictly as a memorial. Christ Himself described the nature - and necessity - of the Eucharist:


  John 6:53 (KJV 1900)
  
  
    Then Jesus said unto them, Verily, verily, I say unto you, Except ye eat the flesh of the Son of man, and drink His blood, ye have no
    life
    in you.
  


I am not aware of any early Christians who maintained that the Eucharist should only be celebrated once a year in lieu of the Passover meal (though I am sure there probably were some).  There are writings, however, of certain Church Fathers explaining the need for frequent (i.e. more than once per year) Communion.  John Cassian (5th c.), a western Church Father with an eastern upbringing wrote:


  We must not avoid communion because we deem ourselves to be sinful. On
  the contrary, we must approach it more often for the healing of the
  soul and the purification of the spirit, to show our humility and
  faith, by considering ourselves unworthy and in need … that we desire
  even more the medicine for our wounds. Otherwise it is impossible to
  receive communion once a year, as certain people do … considering the
  sanctification of heavenly Mysteries as available only to saints. It
  is better to think that by giving us grace, the sacrament makes us
  pure and holy. Such people who receive less often manifest more pride
  than humility … for when they receive, they think of themselves as
  worthy. It is much better if, in humility of heart, knowing that we
  are never worthy of the Holy Mysteries we would receive them every
  Sunday for the healing of our diseases, rather than, blinded by pride,
  think that after one year, receiving once or twice yearly, we become
  worthy of receiving them.
  
  Conferences of John Cassian, Chapter XXI


Similarly, John Chrysostom (4th c.) commenting on those who did not receive the Eucharist at the weekly Liturgy, wrote:


  Look, I entreat you: a royal table is set before you, angels minister
  at that table, the King Himself is there, and yet, you take no account
  of it. Are your garments clean? Then fall down and partake! For
  everyone who does not partakes of the mysteries is standing here in
  shameless falsity. When you behold the curtain drawn, then imagine the
  heavens are let down from above, and that the angels are descending!
  Why stay at liturgy and yet not partake of the table? I am unworthy,
  you say. Then you are also unworthy of that communion you also have in
  prayer. Come!” Holy Father 
  
  Commentary on the Epistle to the Ephesians, Homily III


These writings, although not by the Apostles themselves, might help to explain why the Eucharist was seen by the early Church as something more than an annual replacement of Passover.