Working with a student recently taking her university admission test for Medicine, I have been looking at some BMAT questions. The practice papers with explained answers provide a good understanding of the test format, different question types and skills assessed by the BMAT examination.
There is no doubt that, given the time pressure, students need to be skilled at performing calculations by hand. The fact that questions are in multiple-choice format is helpful, skills in estimation will clearly be an important strategy in answering questions requiring any calculations.
Even though students have to do one GCSE paper without a calculator, calculators are allowed in all A Level Mathematics and Science exams; looking at these BMAT questions made me wonder if we put the calculators aside enough.
This has prompted me to look again at good resources for practising these important skills of estimation and doing calculations by hand.
Estimation by rounding by cparkinson3 is is a complete lesson providing a great variety of problems from basic skills to more challenging ones where learners have to apply their understanding of estimation to solve real-world problems. All resources needed are provided.
On Transum Mathematics you will find many estimation resources including Rough Answers illustrated here. Note the different levels available.
Mathsbot resources give us the opportunity to select non-calculator questions. A revision grid, with such questions, is a good example.
A starter is often a good opportunity to look at strategies for quick calculation, I have used several from Mudd Math Fun Facts which is a site well worth exploring if you are not familiar with it. 
Questions such as this can make a great starter for a lesson and provide the chance to discuss number operations and the relationships between them. Manipulating numbers like this can also help with algebraic manipulation.
Looking for some more examples of this type, I came across a really useful resource on TES, “If I know this then I also know …” by Piers Butler. This would make an ideal lesson starter. As it is an Excel spreadsheet, I thought it would be simple to add another worksheet with the answers and created the Excel file CY If_I_know_this_then_I_also_know_ which is a copy of the original, but just adds another worksheet with the answers.
Thank you Piers! I have added this to the Number resources page.
UK Maths Challenge questions provide an opportunity for working without a calculator. Maths Challenge questions are an excellent resource at any time – not just for preparation for the competitions, particularly with the increased requirement for problem solving skills at all levels. On the outstanding Diagnostic Questions site, you can choose Junior Maths Challenge questions by topic by choosing the Themed Quizzes option. These quizzes consist of sets of four or five questions grouped by topic. (Log in to Diagnostic Questions to use the link.)
Further Resources
Wikipedia mental calculation
Mental Arithmetic Training site
For schools with Sixth Form Students taking this test, I feel it is important that the Mathematics Department has a good understanding of the mathematical skills required. These further resources are useful for understanding the requirements of the test.
Section 1 of the examination is a one-hour test of Aptitude and Skills which includes sections on Problem Solving, Understanding Argument and Data Analysis and Inference. Talking to students it is this section that they find the most demanding. With 35 multiple-choice questions to be answered in 60 minutes without a calculator, students need to work accurately and efficiently.
Section 2 on Scientific Knowledge and Applications includes Mathematics questions, the questions are restricted to material typically included in non-specialist school Science and Mathematics courses. The Mathematics content is covered at GCSE as you can see from the last pages of the BMAT specification. Students will, however, require a level of understanding appropriate for the able target group taking this examination. Questions are in multiple-choice format and calculators may not be used.