They do have social value requirements in most contracts, which includes "a local jobs for local people" element plus the use of SMEs. The problem is that in most contracts its about 10% or at most, and very rarely, 20% of the score that tenderers are assessed on, the majority 50% or more is price, the rest made up of programme and quality of the submission.
What they need to do is accept that costs will be higher but have clauses that prevent significant offshoring of design work. Doing so ensures that the money stays within the UK economy.
Similarly mandating a higher percentage of work being performed by UK based SMEs for both design and construction with more favourable commercial terms (e.g. not contracts that have liabilities of £20m when they are only doing 200k of work) would help. For reference TfGM are a nightmare for doing this and not willing to negotiate more reasonable commercial terms.
Whilst I dont advocate trying to have an insular economy, as we do need international trade, which in turn requires some quid pro quo, we dont give our home grown businesses a leg up, unlike every other country. If you believe the Brexit nutcases, it was European legislation stopping us doing this, which 10 years on is clearly not the case.