From page 5 of House of Commons Library Briefing Paper, Number 01663 (Dec 2016), on Short Money:
Allocations throughout a Parliament are based on the results of the previous General Election.
Quoting from the 1999 resolution which currently governs the system (p37 of the paper):
(a) There are at that time at least two Members of the House who are members of the party and who were elected at the previous General Election after contesting it as candidates for the party, or
(b) There is at that time one such Member who was so elected and the
aggregate of the votes cast in favour of all the party's candidates at that
election was at least 150,000.
So for a new party formed from a split in an existing party, it sounds like these conditions would not be met, as none of the MPs in question would have been elected as candidates for the new party at the last election.
Hence, a new party would receive no money at all until the next general election.
However, the Commons could of course vote to change the rules if it so desired.
EDIT: Regarding the equivalent in the House of Lords (Cranborne money): the Lords resolution of July 2002 (quoted in the above report on pages 43-44) says:
(a) the "Opposition" means the party in opposition to Her Majesty's
Government having the greatest numerical strength in the House of Commons;
(b) the "second largest opposition party" means the party in opposition to Her Majesty's Government (other than the Opposition) with the greatest number of Members of this House among its members;
Although these definitions don't specify when these calculations are made, the mechanism for claiming expenses (as far as I can tell) seems to imply that no change is expected between general elections.
Note that, unlike in the Commons, the Lords simply has a fixed maximum amount that can be claimed by the two largest opposition parties (and also the Convenor of the Cross-Bench Peers).
Note also that the Lords could vary the resolution if circumstances were to require it - and indeed, they did so after the 2010 election to take account of the LibDems being part of the governing coalition, and then again after the 2015 election once that ceased to be the case.