The ECS only provides for submission of a revised Activity Schedule in limited circumstances, it does not contain any rights or obligations in respect of not having an Activity Schedule in the first place, the contract assumes that it has been included by reference in Contract Data part two. I assume this hasn’t taken place as you say that it doesn’t form part of the contract documentation but it might be worth a quick look to confirm.
Assuming Y(UK)2 Housing Grants Construction and Regeneration Act 1996 (as amended) applies the Subcontractor has a right to interim / stage payments unless the contract duration is less than 45 days. Given the contradiction between the contract and legislation the Parties clearly need to find a way resolve this:
Agree a contract variation (exchange of letters may be sufficient but if high risk / value a formal deed of variation or similar provided it complies with clause 12.3).
Agree that the Activity Schedule can be submitted for acceptance under clause 55.3.
Failing this I’d advise the Subcontractor to submit an application for payment based on a reasonable valuation of the works completed. If the Contractor refuses to pay the amount applied for (or some other such some as they determine acting reasonably) it may be possible to adjudicate based on the fact the Contractor is not complying with the Act. The Act will imply certain terms into a construction contract that does not comply with it.
If it’s NEC4 you may want to consider escalation to Senior Representatives to get this resolved to avoid the expense of adjudication in the short term.