Nowadays Royal Assent (Wikipedia) is widely considered to be a formality and in normal circumstances (e.g. no public emergency) its outright refusal would provoke many people to wonder why the monarch has the right or power to veto legislation passed by Parliament.
An exception might be in the case of the Cabinet advising against giving Royal Assent. However, this has not occurred for three centuries.
Queen Anne refused Royal Assent to the Scottish Militia Bill 1708 on the advice of her ministers, who said there was a risk the militia would be disloyal, a risk heightened by the Franco-Jacobite invasion fleet en route to Scotland at the time.
In less fraught circumstances, what would be more likely is quiet words behind-the-scenes about the monarch's discomfort, before this stage were reached, and could something be done to assuage them in some way.
More recently, George V took legal advice as well as ministerial advice on withholding assent to the-then Government of Ireland Bill (aka Irish Home Rule Bill). He decided against refusal without "convincing evidence that it would avert a national disaster, or at least have a tranquillising effect on the distracting conditions of the time."
Of course, given refusal hasn't happened in modern times we can't say what would happen in reality, we can only speculate.
The abdication or death of the monarch doesn't solve the problem because the heir instantly becomes the new monarch. The absence of the monarch doesn't solve the problem because Royal Assent can be approved by certain Counsellors of State.
The monarch can order the Dissolution of Parliament. This would prevent the Bill's passage for some time, particularly if its opponents won a majority. By convention, of course, the monarch only orders Dissolution on advice from their Prime Minster.
Far more likely - but only in the context of the monarch's interests - legislation could be affected much earlier in the process by means of the King's Consent. This is a convention that requires consent when a proposed bill will affect the Crown's prerogatives or interests and there has been criticism that it has been used to allow the monarch to shape legislation. This has a practical advantage in that it avoids the controversy of refusal of Royal Assent by affecting legislation before that stage.
The Guardian published a table of affected Bills alongside its criticism - it's quite varied: Agriculture, Tenant Fees, Fire Safety, Pension Schemes, Data Protection, High Speed Rail, Energy, Small Business and Enterprises, Marine Navigation, Charities, etc.
