No, there is no such law. HL’s stocks, for example, can be sold on HL’s platform, so will Freetrade’s if it IPOs.
It is certainly hard to gauge at this stage what the IPO process will be. Freetrade could be sold to a larger entity, it could liquidate ( this will never happen), it can IPO the normal way by issuing shares (like most companies get into the stock exchange) or it can list directly, which will avoid dilution, the lockup period, but will not raise additional funds.
In the latter two cases, which have a higher likelihood (especially the penultimate one), you will still keep your shares and there is no reason why they should not simply appear on your app. You may be tied up to a lockup period (which may or may not depend on your stake Freetrade), which could delay your ability to sell by up to 6 months.
Of course, I have omitted pesky details since this is a very simplistic overview but there may be additional covenants and conditions by the time Freetrade gets to the IPO stage which will be duly communicated to all investors in due time.