The overall poster you want to adapt will almost certainly be subject to copyright as an artistic work, with certain elements, say the pictures of the leading actors probably having separate copyright if they are photographs, and possibly choice of typeface unless it is very commonplace.
Here's an example of a poster in which nearly all the elements are very distinctive such that copying the main elements would mean a substantial part of the original was being taken, and this would be infringement. Only the title itself, in a different font, would be safe to use. (Image Copyright: Walter Kerr Theater. Used here for the purpose of research under the fair dealing exception s 29 CDPA)
The text on a theatre poster is probably less likely to attract literary copyright, because it will either be factual - the title of the show, names of writer, performers, director etc, dates, address of the theatre - or too short to qualify for copyright. Indeed that last category might include the short snippets quoted from reviews of the show (" 'outstanding drama of the year
' says New York Times"). The fact that their poster can quote these third party reviews is generally evidence that they are too short to attract copyright, or that quoting them is fair use. Obviously you wouldn't want to quote the reviews for the Broadway show, but you might want to say something like, 'Come and see the show the New York critics raved about
' or something similarly cheeky, which would be fine.
So really the thing you have to beware of is copying the original poster too closely. There are only so many ways you can design a poster and you are constrained by how and where the information is displayed on it. So on that basis, you can adopt a similar formula, but avoid re-using too many of the details - background colours, choice of font for the title etc, of the original. Unfortunately there is no magic formula for determining how much needs to be changed for you to be on safe ground.
However I suspect that the mere fact that you want to copy the Broadway poster rather than design your own from scratch is actually the problem. The very fact that you want to create an association between the success of the Broadway show and your own version could amount to passing off, and so if this is the intention it would be better to seek permission to use the format from the producers of the show in the USA.
On a more positive note, there shouldn't be any problem with trade marks, since you have the rights to perform the play, thus you would not be confusing the public as to the origin of the show, the author or playwright being the same for both. And it is unlikely that the Broadway theatre will have registered a trade mark in the UK.