Its not technically true that companies can reclaim VAT. What they can do is offset the VAT they've collected from their customers that they would normally have to send to the government by the amount of VAT they have paid on goods they have bought.
Universities do not sell any VAT chargable goods or services, and so have nothing to offset.
From the HMRC (https://www.gov.uk/vat-businesses, my empahsis:
must charge VAT on their goods or services
may reclaim any VAT they’ve paid on business-related goods or services
If you’re a VAT-registered business you must report to HM Revenue and Customs (HMRC) the amount of VAT you’ve charged and the amount of VAT you’ve paid. This is done through your VAT Return which is usually due every 3 months.
You may want to appoint an agent to deal with HMRC on your behalf.
You must account for VAT on the full value of what you sell, even if you:
receive goods or services instead of money (for example if you take something in part-exchange)
haven’t charged any VAT to the customer - whatever price you charge is treated as including VAT
If you’ve charged more VAT than you’ve paid, you have to pay the difference to HMRC. If you’ve paid more VAT than you’ve charged, you can reclaim the difference from HMRC.
As the amount of VAT "you've charged" is zero, no VAT money changes hands either way between government and universities.
There are exemptions for some, but not all charities. Genreally you have to be selling medical or medical reserach supplies, things related to charitable advertising, and goods/services for disabled people to be allowed to not charge VAT.