This is fine.
You can use initials, shortened names, common nicknames (Bob/Robert), omit middle names, and so forth without causing yourself any problems. Things can get more complicated if you sign by a name that is different from names that you normally use elsewhere - such as if you are called Christopher Smith and you sign as Donald Jones, having not used that name before - but there's no fundamental difference of principle.
One example case is Scott v Soans  102 ER 539, where the defendant John Soans objected to the suit being made against "Jonathan otherwise John Soans". The Lord Chief Justice ruled that "Jonathan otherwise John" could be his name, and that if he'd signed a contract using that name then "what objection could be made to it?" There are several other similar cases from past centuries, some of which may no longer be reliable law since they turn on points of procedure that aren't relevant today, but the general thrust is that if you sign a contract under a certain name, then you can be sued under that name. (And you can sue other people using whatever name you like.)
Mistakes in names can be corrected as part of the general process of contractual interpretation, called "rectification". This more often arises when dealing with company names, say when there are a half dozen closely linked companies with related names, and the issue is which one of them is actually meant to be named; there are some recent cases of this kind, such as Liberty Mercian Ltd v Cuddy Civil Engineering Ltd  EWHC 2688 (TCC). Generally speaking, as Lord Denning said in Nittan v Solent Steel  EWCA Civ J1023-4,
We do not allow people to take advantage of a misnomer when everyone knows what was intended.
Further, the doctrine of "estoppel by convention" means that if you sign a contract under whatever name, then act as if you were bound by the contract, you can't then wriggle out of it on the grounds that the name is not really your own.
In Scots law, which includes certain doctrines imported from Roman civil law, there is a distinction between error in persona and error in nomine. The former means that you were mistaken about who your counterparty really was (such that you wouldn't have made the contract had you known the truth) and the latter means that you had the intended person but made a mistake about their name. The law of error in Scotland is not quite the same as in England and Wales, but in this case it gets to the same basic result: if you agreed on who was to be bound, that's what matters, regardless of the names used.