I don't know if it is common but I have signed my contract on the day I started at my last two jobs. I think you're worrying needlessly about this as I assume certain key points like pay are in your offer letter?
There is always a contract between an employee and employer. You may not have anything in writing, but a contract will still exist. This is because your agreement to work for your employer and your employer’s agreement to pay you for your work forms a contract. Your employer does have to give you a written statement within two months of you starting work. The statement must contain certain terms and conditions.
So legally you don't even have to have a written contract at all, although how it differs from a written statement you'd have to ask a lawyer. I wouldn't imagine any respectable company doesn't give you a proper contract.
Generally, you and your employer can agree to whatever terms you wish to be in the contract, but you cannot agree to a contractual term which gives you less rights than you have under law (see under heading How the rights in an employee’s contract relate to rights in law).
So they can't offer you less than the legal minimum conditions even if you're willing to agree to them.
If the job offer was subject to conditions, for example, the employer needed to take up references or you were required to have a medical, and these were unsatisfactory, you will not be able to make a claim against the employer for any compensation. This is because there was never any contract of employment. There was only a conditional offer of a job and the conditions have not been met.
If the job offer was unconditional, which is unusual, you may be able to claim compensation in the employment tribunal or the county court (Court of Session in Scotland) for breach of contract. This is a breach of contract because you were offered the job with no conditions, you accepted the offer and then the offer was withdrawn.
So if your offer letter contains no clauses you could try to claim compensation from them. However I really think you'd only need to cross that bridge when you came to it.
All quotes were taken from citizens advice. link