The newly engaged pair’s choice of venue and month are expected to be unveiled - the day after they publicly shared their love story.
March or May are the current favourites for the high-profile royal nuptials, but April is thought unlikely.
The Duchess of Cambridge is due to give birth to her third child that month.
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The Queen will also be busy with the Commonwealth Heads of Government Meeting, which is being hosted in London and Windsor in mid-April.
Royal wedding planners in the Lord Chamberlain’s Office at Buckingham Palace are, however, experienced at staging a busy programme of large-scale events.
St George’s Chapel at Windsor Castle is thought a likely, smaller venue for the church wedding - appropriate for fifth-in-line Harry.
But the prince and Ms Markle may decide to opt for a full-scale fairy tale ceremony instead and choose Westminster Abbey, where William and Kate wed, or St Paul’s Cathedral where Harry’s parents married.
The couple received more well wishes on Tuesday from the Duke and Duchess of Cambridge.
On a visit to the Foundling Museum in central London, the duchess said she and William were “absolutely thrilled”, adding: “We wish them all the best and hope they enjoy this happy moment.”
Harry revealed on Monday in a television interview to celebrate their engagement how “the stars were aligned” when he fell for the American actress.
The couple, who have been dating for just 16 months, told how they met on a blind date set up by a mutual friend.
Former Suits star Ms Markle told how she could not wait to say “yes” to the prince when Harry got down on one knee a few weeks ago as they cooked a roast chicken dinner at his home, Nottingham Cottage at Kensington Palace.
Indicating the depths of his feeling for his bride-to-be, the prince said: “The fact that I fell in love with Meghan so incredibly quickly was sort of confirmation to me that everything, all the stars were aligned, everything was just perfect.
“It was this beautiful woman just sort of literally tripped and fell into my life, I fell into her life.”
A few months into the relationship, Harry said he had to have “some pretty frank conversations” with his girlfriend about how her life may change when she entered the royal family.
He added: “But I know that at the end of the day she chooses me, and I choose her, and therefore whatever we have to tackle together or individually we’ll always be us together as a team, so I think she’s capable ... she’s capable of anything.”
The couple told the BBC’s Mishal Husain how Ms Markle had met the Queen twice, an individual she described as an “incredible woman”.
Ms Markle was also a big hit with the Queen’s beloved corgis, with the prince joking: “I’ve spent the last 33 years being barked at; this one walks in, absolutely nothing ...”
Harry praised the support of the pair’s families and said he believed his bride-to be and mother Diana, Princess of Wales would have been “thick as thieves, without question”.
Ms Markle’s engagement ring features diamonds from Diana’s personal collection, and a stone from one of their favourite places, Botswana.
Ms Markle will become an HRH and a senior royal after marrying Harry and joining the Windsors, and is expected to become a duchess, as Kate Middleton did when she became the Duchess of Cambridge after saying her vows in 2011.