A little clarification if I may....
Say work stopped 1st March due to soft spot. Critical progress stopped for 3 weeks. EOT sought for 3 weeks based in likely effect of the delaying event. [The Architect must award this (or another period if there are other factors) within 12 weeks [JCT05]]
The extra prelims are the additional costs incurred as a result of the event at the time. This is important. It is not necessary to use a formula for extended prelims if the actual cost can be calculated.
eg prelims are £120,000 for a 20 week project, making £6000 per week. It is tempting to say that the prolongation cost of the event is 3 weeks x £6000 = £18000. This is not necessarily correct. £6000 is only the average rate.
The question is, what were the additional expenses in the 3 weeks from 1st March? You could have an setting-out engineer at £2000 per week and excavators stood idle at £2500 per week. These wouldnt be there later in the project. Add these to the other prelim costs and the 3 week period in March might cost much more than £18,000.
The cost of the piling will not be determined by BQ rates as the Bill won't contain any - it wasn't part of the scope.
The value in this case will be the nett cost plus an allowance for overheads and profit.
Generally, there are 4 ways of valuing varations, in this order....
1) Bill rates
2) Rates derived therefrom
3) Market value
4) Daywork (or cost-plus)
So heres the buildup of the final value
Credit for strip foundations not installed - (Bill rates and measure)
Piling (Cost plus)
Standing time for operatives / plant (daywork)
Prolongation (cost in that 3 week period) NO Profit - Prolongation cost is a restitution, not a value - only costs recovered
Clear as mud, yes?