The law on registration under what is known as the”forward look” says:
“a person becomes liable to be registered if, at any time, there are reasonable grounds for believing that the value of his taxable supplies in the period of 30 days then beginning will exceed the registration threshold.”
At this moment you don’t know whether you will get this work or not, so cannot be required to register due to it.
But – if you are successful, there will be a point in time at which you have been awarded the contract, but have not yet supplied the service. Assuming that the £120k has a single “tax point” (more on this later…) at that point you will know that you will exceed the threshold, and so be required to register then. The full £120k would then be liable to VAT.
Neither exception nor exemption would apply. These are technically different terms. Exception can only apply to registrations required on the “backward look” (last 12 months’ turnover), and is generally where the supplier would be entitled to deregister immediately. Exemption is where your taxable turnover exceed the threshold due to zero rated supplies.
There’s the practical aspect about how you deal with your supplier. Are they a business that would be entitled to reclaim any VAT charged on their VAT returns? If so, explain to them that the contract would require you to VAT register. You could agree to bill them for £144k, and then issue a receipted VAT invoice (which would allow them to recover the VAT) as soon as your registration number came through.
Another possibility regarding tax point. You mention that you’re supplying software as a service. Would this £120k cover use of the software over a period of time? If so, consider invoicing periods. The tax point of a “continuous supply of services” is generally the date of each invoice (see HMRC notice 700 paragraph 14.3). So if it’s use of the software for a year, could you agree in the contract to (say) invoice £30k per quarter? You’re then not required to register under the “forward look”, as you’ll only be getting £30k plus your other taxable turnover in the next 30 days. You will only need to register under the “backward look” once the previous 12 months’ taxable turnover exceeds the £85k. Based on your figures, you might not reach that point until the third quarter, by which stage there will only be one quarter left liable to VAT.